Pramana - J. Phys., Vol. 39, No. 5, November 1992, pp. 413-491. © Printed in India.
sdg Interacting boson model: hexadecupole degree of freedom in nuclear structure t
Y DURGA DEVI and V KRISHNA BRAHMAM KOTA Physical Research Laboratory, Ahmedabad 380009, India
MS received 13 May 1992; revised 27 August 1992 Abstract. The sdg interacting boson model (sdglBM), which includes monopole (s), quadrupole (d) and hexadecupole (g) degrees of freedom, enables one to analyze hexadecupole (E4) properties of atomic nuclei. Various aspects of the model, both analytical and numerical, are reviewed emphasizing the symmetry structures involved. A large number of examples are given to provide understanding and tests, and to demonstrate the predictiveness of the sdg model. Extensions of the model to include proton-neutron degrees of freedom and fermion degrees of freedom (appropriate for odd mass nuclei) are briefly described. A comprehensive account of sdglBM analysis of all the existing data on hexadecupole observables (mainly in the rare-earth region) is presented, including l/4 (hexadecupole deformation) systematics, B(IS4; 0~s~4~+) systematics that give information about hexa'decupole component in 7-vilaratlon, E4 matrix elements involving few low-lying 4 + levels, E4 strength distributions and hexadecupole vibrational bands in deformed nuclei. Keywords. Interacting boson model; IBM; g-bosons; sdglBM; dynamical symmetries; collective modes; SU(3); coherent states; spectroscopy; mappings; 1/N method; proton-neutron IBM; F-spin; IBM-2 to IBM-1 projection; interacting boson-fermion model; scissors states; pseudo-spin; hexadeeupole deformation; E4 strength distributions; hexadecupole vibrational bands. PACS Nos 21-10; 21.60; 23.20; 27.70
Table of Contents
1. Introduction 2. sdg Interacting boson model 2.1 Dynamical symmetries 2.2 Geometric shapes 2.3 Two-nucleon transfer 2.4 Hexadecupole vibrational bands 2.5 sdg Hamiltonians 2.6 Spectroscopy in sdg space 2.7 Studies based on intrinsic states 3. Extensions of sdglBM 3.1 Proton-neutron sdglBM 3.2 sdg Interacting boson-fermion model 4. E4 Observables and sdglBM 4.1 [/4 Systematics tThe survey of literature for this review was concluded in December 1991. 413 414 Y Durga Devi and V Krishna Brahmam Kota
4.2 B(IS4; 0 GS+ ~4 ~,+) systematics 4.3 Select E4 matrix elements 4.4 E4 Strength distributions 4.5 Excited rotational bands 5. Conclusions and future outlook
1. Introduction
Atomic nuclei exhibit a wide variety of collective phenomena, and their study initiated by Rainwater, Bohr and Mottelson (Rainwater 1950; Bohr 1951, 1952; Bohr and Mottelson 1953, 1975) continues to be a subject of intense investigations. The observation of enhanced (with respect to the extreme single particle model) quadrupole moments, and electric quadrupole (E2) transition strengths and the appearance of: (i)the one phonon 2 + state and the two phonon (0÷,2+,4 +) triplet states characterizing the quadrupole vibrations; (ii) the rotational spectrum characteristic of a quadrupole deformed (spheroid which is axially symmetric) object; (iii) the spectra that exhibit the so-called v-unstable motion (V is the axial asymmetry parameter), in several nuclei establish the dominance of the quadrupole collective mode in the low-lying states of medium and heavy mass nuclei; the equilibrium shapes that correspond to (i)-(iii) are spherical, spheroidal and V-soft (or ellipsoidal), respectively. Bohr and Mottelson (1975) developed the collective models (which are geometric in origin) to explain and predict the various consequences of this collective mode by seeking a multipole expansion of the nuclear surface, and treating the resulting expansion variables as collective (canonical) variables (i.e. the radius R(0,~b) is expanded as R(O,~)=Ro{I+~',~t~ Yx.(O,~)} ~>2,1~1~2 where etzu are collective variables). In the last 15 years there has been a significant amount of activity devoted to exploring other collective modes; octupole (Leander and Sheline 1984; Rohozinski 1988; Leander and Chen 1988; Kusnezov 1988, 1990), hexadecupole (see §4 ahead), scissors (Dieperink and Wenes 1985; Richter 1988, 1991) and collective states which include or-clusters (Daley and Iachello 1986), two-phonon (Borner et at 1991) and two quasi-particle states (Solov'ev and Shirikova 1981, 1989; Nesterenko et al 1986; Solov'ev 1991) etc. Figure la illustrates some of the nuclear shapes and, as an example, figure lb shows the dynamics associated with quadrupole shapes. Although the collective models were enormously successful in explaining data (Bohr and Mottelson 1975) and predicted many new phenomena in nuclei (the recent observation and exploration of super deformed bands (Twin 1988; Janssens and Khoo 1991) is a good example; super deformed shapes are prolate ellipsoids with axes ratios of 2:1:1 and they are first discovered in 15ZDy) nuclear physicists time and again tried to develop a model which in some way incorporates the microscopic (shell model) aspects of nuclei and at the same time retains the simplicity and the important features of the Bohr and Mottelson model. The outcome of this activity is the interacting boson model (IBM), which was proposed by Arima and Iachello (1975, 1978a) in 1975. Sphere Prolate Spheroid Oblate Spheroid Octupole (0,0,0) (0.3,0,0) (-0.3,0,0) (0.0.3,0)
Hexadecupole Hexadecupole Quadrupole -t- Hexad{,cupoh, Quadrupole -F Hexadecupoh" (0,0,0.15) (0,0,-0.15) (0.3,0,0.15) (0.3,0,-0.15) Figure l(a). Some typical axially symmetric nuclear shapes. Corresponding to each shape, given are the parameters (fl2,fla,fl4) that define the nuclear mrface: R(O, dp)= Ro[1 + f12 Yo(O, 2 d?) + f13 Y0 3 (0, ~b) + fla Y0 4 (0, ~b)]. 416 Y Durga Devi and V Krishna Brahmam Kota
(b)
VIBRATION ROTATION SCISSORS CLUSTERING
Figure 1(b). Collective modes associated with quadrupole shape.
The interacting boson model provides an algebraic description of the quadrupole collective properties of low-lying states in nuclei in terms of a system of interacting bosons. The bosons are assumed to be made up of correlated pairs of valence nucleons [as Balantekin et al (1981) put it "the correlations are so large that the bosons effectively lose the memory of beingfermion pairs---"] and they carry angular momentum f = 0 (s-boson representing pairing degree of freedom) or • = 2 (d-boson representing quadrupole degree of freedom). The bosons are allowed to interact via (1 + 2)-body forces and the boson number N (which is half the number of valence nucleons; particles or holes) is assumed to be conserved. This model is remarkably successful in explaining experimental data (Iachello and Arima 1987; Casten and Warner 1988). In the words of Feshbach (Iachello 1981) "IBM has yielded new insights into the behaviour of low-lying levels and indeed has generated a renaissance of the field of nuclear spectroscopy". One of the most important features of IBM is that it has a rich group structure. This model admits three (only three) dynamical symmetries (Arima and Iachello 1976, 1978b, 1979) denoted by the groups U(5), SU(3) and 0(6) and they describe the vibrational, axially symmetric deformed and 7-unstable nuclei, respectively. Apart from these limiting situations, the model allows for rapid analysis of experimental data in the transitional nuclei (by interpolating the limiting situations (Scholten et al 1978; Casten and Cizewski 1978, 1984; Stachel et al 1982); for example 628m, 76Os, 44Ru isotopes interpolate I-U(5), SU(3)], [SU(3), O(6)-I and [U(5), 0(6)] respectively) as the general IBM hamiltonian possesses only seven free parameters and the model-space dimensions are small (~ 100 for N ~< 15). Some of the recent reviews on the subject (Arima and Iachello 1981, 1984; Iachello and Arima 1987; Casten and Warner 1988; Dieperink and Wenes 1985; Lipas et al 1990; Iachello and Talmi 1987; Klein and Marshalek 1991; Elliott 1985; Kusnezov 1988; Kota 1987; Vervier 1987; Iachello and Van Isacker 1991; Iachello 1991) give: (i) details of the analytical results derived in the three symmetry limits and the geometric corres- pondence brought via coherent states (CS); (ii) the remarkable success in correlating observed data all across the periodic table (A/> 100); (iii) extension to proton-neutron IBM (p - n IBM or IBM-2 and the IBM where no distinction is made between protons and neutrons is often called IBM-1 or simply IBM) which gives rise to the concept of F-spin and describes the scissors I ÷ states; (iv) microscopic (shell model) basis of sd9 Interacting boson model 417
IBM; (v) IBM-3, 4 models which incorporate isospin (they allow one to study A = 20- 80 nuclei); (vi) spdf (or sdf) boson model to describe octupole collective states; (vii) extension to odd-mass nuclei via interacting boson fermion model (IBFM) where the single particle (fermion) degree of freedom is coupled to the collective (boson) degrees of freedom of the even-even core nucleus; (viii) extensions of the model to include two, three and multi-quasi particle excitations for describing high spin states, odd-odd nuclei etc. In the first two workshops (held in 1979 and 1980 respectively; Iachello 1979, 1981) that recorded the success of IBM, in his concluding remarks Feshbach raised the following questions: "the success of the interacting boson model would indicate a whole new method for looking for symmetries. As an example, are there 4 ÷ bosons and if so, what kind of symmetry is implied? In addition to the interaction among the 4 ÷ bosons, for example the 0 ÷ and 2 ÷ subspaces, what kind of broken symmetry is implied by the existence of the 4+? Comparison with the experimental data would obviously help to determine the impact of the 4 + boson .... The question we now ask has correspondingly changed. We no longer ask if the s and d boson description is adequate. The question has become: What is the range of phenomena for which only s and d bosons are needed?.., for even-even nuclei, indicates that including the g(~ = 4 ÷) boson can on occasion be advantageous as noted by Barrett, Gelberg and Van lsacker. Turning now to microscopic theory, that is theory which attempts to relate the IBM to a more fundamental underlying theory .... the more fundamental theory is for most part the nuclear shell model .... In the microscopic theories, will it be necessary to include the E = 4 ÷ boson?". These questions and remarks together with the fact that in the past few years there is accumulation (see § 4) of considerable amount of experimental data on E4-matrix elements and strength distributions on one hand and the importance of G(L ~ = 4 ÷) pairs as brought out by the microscopic theories on the other hand lead several research groups to explore the extended sdg interacting boson model in detail (we refer to it as sdglBM or sdglBM-1 or simply glBM and the sdlBM is often referred to as IBM). Experimental data on hexadecupole deformation parameter /~4 (it gives B(E4; 0Gs~41+ 4- )) all across the rare-earth region are available for many years. However, only in the last ten years different types of E4 data (mainly in rare-earths) are obtained and they include (§ 4 gives details): (i)/~4 data for several actinide nuclei; (ii) isoscalar mass transition density B(IS4; 0gs ~ 4+ ), which gives information on Y42 deformation (~42 in R(O, qb) expansion), all across the rare-earth region; for many purposes B(IS4) can be treated as B(E4); (iii) E4 transition matrix elements involving 4 ÷ (i ~< 6) states in some of the Cd, Pd, Er, Yb, Os and Pt isotopes; (iv) E4 strength distributions in xX2Cd, l S°Nd and 156Gd; (v) hexadecupole transition densities for exciting some of
the 4 ÷ levels in Cd, Pd, Os and Pt isotopes; (vi) K s - 03 ÷ , 31 + , 22 + and 4 ÷ bands which can be interpreted as bands built on hexadecupole vibrations. The sdglBM with the hexadecupole (f = 4) g-bosons is ideally suited for analyzing the above data, which in turn provides stringent experimental tests of the model. It is needless to state that the data analysis provides understanding of the role of the E4 degree of freedom in nuclei. The situation with regard to data on E4 observables is well summarized by Walker et al (1982): "Single phonon shape oscillations in deformed nuclei have been the subject of extensive investigations. Much has been learnt about the systematic occurrence of the quadrupole (~, ~) and octupole modes. The situation for the 418 Y Durga Devi and V Krishna Brahmam Kota hexadecupole mode is one of comparative ignorance though some nuclei received some attention ...". In addition to the direct evidence provided by E4-observables, the microscopic theories of IBM provide a strong basis for the inclusion of g-bosons in IBM. In the spherical basis, Otsuka (1981a, b) using the so-called favoured pairs, Scholten (1983) using seniority scheme in multi j-shell, Van Egmond and Allaart (1983, 1984) and Allaart et al (1988) using the broken pair method and Halse et al (1989) and Yoshinaga (1989a, b) using shell model basis in a single j-shell demonstrated that pairs coupled to angular-momentum 4 + (G-pairs) in addition to the S(L"= 0 + ) and D(L"= 2 + ) pairs are essential (and sufficient) for a proper description of moment of inertia, binding energies etc. It should be pointed out that the S, D and G pairs are microscopic (fermionic) counterparts of the s, d and g bosons respectively. Similarly in the deformed basis (Bohr and Mottelson 1980, 1982; Yoshinaga et al 1984; also Bes et al 1982; Otsuka et al 1982) using Nilsson-BCS basis, Pannert et al (1985) using Hartree-Fock-Bogoliubov (HFB) intrinsic states and Dobaczewski and Sskalski (1988, 1989) employing Skyrme interaction and using the HFB method showed that the composition of the intrinsic states (that define the rotational bands) involve about 90~ of S and D pairs and 10~ G pairs but at the same time G-pairs are shown to be essential for the proper description of the above mentioned observables (binding energies, quadrupole moments, moment of inertia etc.). The conclusions from these studies is as stated by Yoshinaga et al (1984) "the inclusion of hexadecupole degree of freedom in addition to the monopole and quadrupole degrees of freedom is important and sufficient to reproduce physical quantities". Finally it should be added that there are a large number of other signatures indicating that g-bosons should be included in the IBM (Casten and Warner 1988; Devi and Kota 1990a). For instance: (i) g-factor variation with respect to angular momentum as seen in 166Er (Kuyucak and Morrison 1987a); (ii) anharmonicity associated with the K" = 4 + bands in 168Er (Yoshinaga et al 1986); (iii) extension of band-cut-off beyond sdlBM value (Dukelsky et al 1983); (iv) macroscopic SU(3) scheme (Chakraborty et al 1981; also Bhatt et al 1974) indicating that the ground state rotational band is closer to (4N,0) SU(3) irreducible representation (irrep) of sdglBM than to the (2N,0) irrep of sdlBM; (v) the occurrence of levels or bands (in several nuclei) that lie outside sd boson space (Akiyama 1985; Devi et al 1989; Devi and Kota 1991a) etc. In view of the above discussion it is clear that the sdglBM should be explored, understood and applied and the progress made in this direction is reviewed in this article. It should be pointed out that preliminary accounts of sdglBM are given in the articles by Casten and Warner (1988) and Devi and Kota (1990a). Our purpose here is two fold (the two parts of the title reflect the dual purpose of the article): (i) to give a coherent view of the developments in sdglBM; (ii) to bring to one place the available E4 data (most of it has been taken in the last 6-8 years) and its analysis in terms of sdglBM. To the authors knowledge this is the first article to focus exclusively on the E4 properties of nuclei. It is important to mention that one can employ the microscopic HF and HFB methods (Goodman 1979; Svenne 1979; and Libert and Quentin 1982), Kumar's dynamic deformation model (DDM) (Lange et al 1982j, Rowe's (1985) symplectic Sp(6, R) model, Draayer's (1991; see also Ratnaraju et al 1973; Draayer and Weeks 1984; Castanos et al 1987) pseudo-SU(3) and pseudo- symplectic models, quasi-particle phonon nuclear model (QPNM) (Solov'ev 1991), multiphonon model (MPM) (Piepenbring and Jammari 1988; Jammari and Piepenbring sdg Interacting boson model 419
1990), Nilsson-Strutinsky method (Nilsson et al 1969) etc. to describe some (in some cases all) of the E4 data. As so far none of the models are used in analyzing all the different types of E4 data mentioned above, we restrict our discussion to sdgIBM and wherever appropriate the results from other models are shown for comparison. Moreover, although there is E4 data for mass A < 100 nuclei (Endt 1979a, b; Matsuki et al 1986; Fujita et al 1989), they are not dealt with in this article, since the data is sparse and since isospin should be included in sdgIBM for dealing with A < 100 nuclei, an extension which is not yet available (see however Zuffi et al (1986), Chen et al (1986, 1988) and Devi et al (1989)). Now we will give a'preview. This article is divided into two parts. The first part describes the sdoIBM (§2) and its extensions (§ 3) and the second part (§ 4) presents the sdgIBM analysis of E4 data (mostly for rare-earth nuclei). Technical details are kept to a minimum throughout the article. The sdgIBM admits seven dynamical symmetries denoted by the groups SU do(3), SU~dg(5), SU~do(6),O~do(15), U~d(6)~ U0(9), Us(1)~) Un0(14) and Ud(5)~ Uso(10). Section 2.1 gives the generators, basis states and Casimir operators in the various symmetry limits. The geometric shapes corresponding to the seven dynamical symmetries and also the general sdgIBM hamiltonian are studied using CS and the results are given in § 2.2. Two nucleon transfer (TNT) reaction is an important probe in providing information about shapes and shape phase transitions and thereby determine the relevance ofsdgIBM symmetries. Section 2.3 deals with TNT in sdgIBM framework. Coupling a 9-boson to the Un(5), SUsd(3) and O d(6) limits of sdIBM generates hexadecupole vibrational levels/bands in vibrational, rotational and v-unstable nuclei respectively. Classification of these band structures in these three limits is briefly discussed in §2.4. As the limiting situations described in §§2.1-2.4 are not, in general, adequate for dealing with the available E4 data, it should be clear that detailed (numerical) spectroscopic calculations in sdo space are called for. To this end, it is essential to have adequate knowledge of interactions in the sd9 space. Several sd9 hamiltonians are constructed using phenomenological (or symmetry) considerations and microscopic (shell model based) theories, which are described in- § 2.5. Detailed spectroscopic calculations (spectra, E2, E4 and M 1 transition strengths, occupation numbers, (= 0 TNT strengths), based on matrix diagonalization are carried out for several rare-earth nuclei, including chains of isotopes. The different bases employed and some of the results obtained using the same are given in § 2.6. For well-deformed nuclei it is appropriate to deal with intrinsic states with or without angular momentum projection. This gives rise to (i) numerical Hartree-Bose (HB) plus Tamm-Dancoff Approximation (TDA) method and (ii) 1/N expansion method which gives analytical expressions accurate to order 1/N for spectra, transition moments etc.; Section 2.7 deals with this topic. Inclusion of proton-neutron degrees of freedom in sdoIBM gives rise to p - n sdgIBM (or sdoIBM-2) and similarly coupling of an odd fermion to sdgIBM core gives rise to sdo interacting boson fermion model (sdgIBFM) for odd mass nuclei. Details of these two important extensions of sdgIBM are given in §§3.1, 3.2 respectively. The /~4 systematics in rare-earth and actinide nuclei are studied employing the SUed0(3) and SU ,g(5) CS together with mapped (single-j-shell mapping in proton and neutron boson spaces) and sdgIBM-2 to sdoIBM-1 projected hexadecupole transition operator and the results are given in § 4.1. The above procedure with more elaborate mappings (multi-j-shell) of the E4 operator describes the recently observed B(IS4; 0~s~4 +) systematics in rare-earth nuclei and the results are given in § 4.2. Detailed spectroscopic calculations employing 420 Y Duroa Devi and V Krishna Brahmam Kota the various sdg hamiltonians and E4 operators are carried out for E4 matrix elements in Zl°Cd, 1°4-11°Pd, 168Er, z72yb, 192Os and 194-x98pt isotopes and they are described in §4.3. E4 distributions are available only for lZ2Cd, 15°Nd and Z56Gd. Calculations using matrix diagonalization for (~Z2Cd, 15°Nd) and HB + TDA for (z.~ONd' Z56Gd) are carried out. These results together with predictions for 152.zS4Sm are reported in §4.4. The excited rotational bands with K"= 0~-, 2 2 , 31 and 4 + (in addition to #, 7 bands) are observed throughout the rate earth region and these bands in some of the nuclei are purely hexadecupole vibrational in nature and they have a simple description in the SUu(3) x 10 limit. In the situation where there are two phonon admixtures they are described using either SUsdg(3) limit or mean field methods. The structure of these excited rotational bands as determined by the above descriptions in sd0IBM are summarized in §4.5. Finally § 5 gives some concluding remarks and future outlook.
2. sdg Interacting boson model
In the sdglBM, the low-lying (quadrupole + hexadecupole) collective states of a nucleus are generated as states of a system of N interacting bosons occupying levels with angular momentum t = 0 (s-boson), t = 2 (d-boson) and E = 4 (g-boson) (see figure 2). All the states of a N-boson system belong to the totally symmetric irrep {N} of U(15) group where U(15) is the unitary group in 15 dimensions. Here the 15 dimensions correspond to the 15 single particle states which is the sum of one state coming from 's', five from 'd' and nine from 'g' orbit respectively. Two particle states in sdg space are is2 L=OML>, lsdL=2ML), IsoL=4Mc>, Id2 L=O, 2,4ML), Idg L = 2, 3, 4, 5, 6 ML) and Ig 2 L = 0, 2, 4, 6, 8 ML). With this, the general 1 + 2 body hamiltonian (HareM) which preserves angular momentum and conserves the boson number in the sd# space, contains 35 free parameters; three single particle energies (SPE) which are denoted by es, ed and e9 and thirty two two-body matrix elements (TBME) denoted by VL(.... ) where for example VL(sdg9)= <(sd)LI Vl(og)L> and similarly all other TBME are defined. In terms of the s, d and g boson number
~g x x g (£ :41 m t :4,3,2,1,0 9 - I,-2,-3,-4)
E d d (/. :2; m/. =2,1,0,-1,2) .5
E s s(£:O~ m t :0) I
15
Figure 2. The configuration sSd3g2 of sdo Ix)son system. The single boson energies corresponding to the spherical s, d and g are denoted by e,, e~ and e0 respectively. There are total of 15 single boson states and the corresponding (~',m/) quantum numbers are also given. sdg lnteraetin 0 boson model 421 operators ~,, fldand ~g, the creation operators Sot, d*~and O*~, and destruction operators So(go = So), d,.(d,, = (- l)"d-m) and 9,,,(9,, = (- 1)"'9_m,) respectively, the explitic form of HgmM is
H#B M = es~ s + ed~ d -F egl~g + VO(ssss) [(stst)°(~s')°] ° + ½ V°(ssd d) [(sts*)°(dd) ° + h.c.] ° + ½ V°(ssgg)[(s'st)°(99) ° + h.c.] °
+ ,,/~V~(sddo)[(stdb~(2~) ~ + h.c.] °
+(:)UZV2(sdgo)[(s*d*)Z(99)2+h.c.]°+3V'*(soso)[(s*o*)4(gO)4]°
3 + -- V4(sgdd)[(sto*)'*(~ 4 + h.c.] °
+ 3v"(sada)[(sM)'(J9)" + h.c.3 0 + ~ 3 V 4 (saaa)[(sM) 4 (99) 4 + b.c.3 °
+ ~ ½(2Lo + 1) '/2 VL°(dddd)[(d*dt)L°(d~l)L°] 0 Lo = 0,2,4
+ ~ [(2Lo + 1)/2] '/2 VL°(dddo)[(dtdt)L°(dg)L°+h.c.] ° Lo=2,4
+ ~ ½(2Lo + 1) '/2 vLo(ddgg)E(cl*d*)Lo(99)Lo + h.c.] ° Lo=0,2,4
+ ~ (2Lo + 1) '/2 vLo(dgdg)[(dtgt)Lo(dg)L°]° Lo= 2,3,4,5,6
+ ~ [(2Lo + 1)/2] '/2 VL°(dggg)[(d*g*)L°(90) L° + h.c.-I ° Lo= 2,4,6
+ ~ ½(2L o+1) '/2V Lo (gggg)[(gg)t ? Lo (99) Lo ] 0 • (1) Lo = 0,2,4,6,8 In (1), I-( - - )Lo( _ _)L o-I0 is the standard angular momentum coupled tensor (Edmonds 1974). In the first approximation, as in sdlBM, it is assumed that the transition operators in sdglBM are one-body operators. Explicit forms for electromagnetic operators (magnetic dipole (M1), electric quadrupole (E2) and electric hexadecupole (E4)) are
T u' = x/z~[gd(dt~) ' + x//-6gg(g*9) ' ], (2) Tr2 = e(o~(sty+ d*s")2 + e(2)/dtd)2+22,, e(2)¢dta2,, ~ + ~"*~)2, + e(,,~(g*9)2, (3) T E4 = e~o~(S*9 + g's3 4 + %2(d(4) ?~4d) + e:4(4)( d*g~ + g ?~4d) + e4,,(g(4) t~4g). (4)
In (2), gd and go are g-factors and in (3, 4) e(.-)_ 's are effective charges. The operators 422 Y Durga Devi and V Krishna Brahmam Kota
T r2 and T r4 are often referred to as Q2 and Q4, respectively. Besides transition moments, the other observables of interest are isomer (6(~2)) and isotope (A(~2)) shifts and L=0TNTstrengths(P~°; (+) for addition and (-) for removal, p = n(v) for protons (neutrons)). The corresponding operators are
= <¢2 + ') -
=7, + ~2 [ (¢id)(~ +' -- (ttd)(0~)] + 73 [ (fl,)(~ +') -- (ti0)(0~)], (6)
^ -IX/2
pe~0= ¢ 7 '/2 _ __o[n _/"~"- ~-/~" (ad +,.)] (7)
In (5, 6) Yx, 72 and Y3 are free parameters,
N+ 14"/(2L+ 1) (L+ 1/2)2 O(N,L)= 14 )~exp 2o.2 ,
= 14 } = N(N+15). (8)
For example from (8), D(8,8)= 1405, D(12, 10)= 24555 and D(14,4)= 51258, while the exact values are 1460, 25008 and 53748 respectively. Broadly speaking, three different approaches are adopted in literature to take into account g-boson effects. The first method is to use some renormalization technique so that one can still employ the sd boson space; this method fails for states that lie outside sd space such as the K" = 3~ band in XbSEr, 43 state in 192Os, (6~-, 8~-) states in 2°Ne etc. The second method is to allow for weak coupling which is equivalent to performing calculations in a truncated space consisting of (sd)N, (sd)N-X(g) 1, ..., (sd)N-"7"(g)"7 "" configurations with the maximum #-boson number ngmax taken to be small (~ 2-3). The third method which corresponds to strong coupling is to treat the #-bosons on equal footing with s and d-bosons. Studies based on dynamical symmetries and those involving intrinsic states belong to this category. The first method is ill suited for studying E4 properties of nuclei and hence it is not discussed in this article; see.however Sage and Barrett (1980), Druce et al (1987), Otsuka and Ginocchio (1985) and Amos et al (1989). As Amos et al (1989) state "the Otsuka and sdg Interacting boson model 423
Ginocchio (1985) renormalization model provides a qualitative guide to gross structure of sdg boson system .... explicit inclusion of a few g-bosons in any shell model calculation seems less fraught with uncertainties." The weak coupling approach which is essentially numerical in nature, but appropriate (derived from the success of sdlBM) for a majority of nuclei is discussed in § 2.6 and the strong coupling approach which is essentially analytical in nature is dealt with in §§2.1-2.4 and 2.7. Let us begin with the dynamical symmetries of sdglBM.
2.1 Dynamical symmetries The Hgm~a in (1) is solvable when the free parameters gs and yL( .... ) take a particular set of values so that it becomes a linear combination of the Casimir operators of the various groups in a group sub-group chain U(15) ~ G = G' =... ~ 0(3) (9) where 0(3) is the group corresponding to angular momentum in sdg space. Then H#mM is said to possess a dynamical symmetry, which is denoted by the first sub-group (G) in the given chain. A complete classification of the dynamical symmetries and their algebra exhaust all the analytical (limiting) solutions of glBM. Kota (1984) and Meyer et al (1986) showed (the former using physical arguments and the latter using representation theory) that glBM possesses seven dynamical symmetries and they correspond to four strong coupling limits SUsdg(3), SU dg(5), SU dg(6) and O dg(15) and three weak coupling limits U d(6)~Ug(9), Udg(14) and Ud(5)~U~(10) respec- tively; initial studies on the SU ~(3), SU,a~(5) and U~(14) limits are due to Ratnaraju (1981), Kota (1982) and Wu (1982), Sun et al (1983), and Ling (1983) respectively. The complete group chains and the corresponding group labels or irreps are given in figure 3. The group generators, irrep labels (quantum numbers), quadratic Casimir
Usdg(15)----.~
{ml 2m3 "}j {nsd'ns't'nd} j {"g}- / ~/ "~ ~ ~J ,SU.A(3) Osd(6) Ud(5)', / ~ ~ w ~~
j \ o. ~5~,, [,g] ,i\ ~, ,/ I,.l ods) ) Od{3) Lg 5p~ (6)
" ...... (D I <"'"'> Odgh ",](5) 0~3 1' ),
Figure 3. Dynamicalsymmetry group chains in sdglBM. The dashed box gives the group chains in sdIBM. The irrep labels correspondingto each group in the various symmetry chains are also shown. The subscript labels (sdg, sd, d,...) definethe space in whicha given group is realized. 424 Y Durga Devi and V Krishna Brahmam Kota operators (C2(G); G denoting a group) and their eigenvalues are worked out in detail by Kota et al (1987). It should be pointed out that the SU~a0(3) group chain follows from the classic work of Elliott (1958a, b) as the ~ = 0, 2 and 4 values corresponding to the s, d and g bosons can be viewed as arising out of the oscillator shell with principle quantum number X = 4, the SU dg(5) group chain by recognizing that a particle carrying angular momentum ~' = 0, 2 and 4 can be thought of as composed of two pseudo-bosons each carrying angular momentum 7 = 2, similarly the SU,ag(6 ) chain with two pseudo-fermions each carrying angular momentum j'= 5/2 and finally the O,dg(15) chain corresponds to generalized seniority in sdg space (with zero coupled pair S+ = 15 -1/2 (s*.st+ dt.d*+ gt.g,)). The remaining three chains U a(6)@U0(9), Uag(14) and Ua (5) ~ U~g(10) can be realized by demanding that { n~a = n + n a, ng }, {nag = nd+ n o, n,} and {n g = n~ +ng, rid} respectively to be good quantum numbers.
SU~do(3 ) limit: The group chain in the SU~do(3) limit is U do(15) D SU,d~(3 ) D Od.(3 ). The SU~ag(3) group is generated by the eigh[ operators {QZis), L} where Q2(s)is'the quadrupole generator and L is the angular-momentum O. (3) generator. The explicit form of Q2(s) is obtained by taking the matrix elements o°f r 2 y2(O, c~) in the X = 4 oscillator shell and similarly L operator is defined. The result is,
Q](s) = 4(7/15)l/Z(st d + d* s")2u - 11 (2/21)l/e(d* d)~ + 36/(105)~/2(d* 0 + g* d).~2 - 2(33/7)1/2(gt~) 2 (10)
The factor x/~ in (11) is chosen so that the L.L matrix elements will be L(L+ 1). Using the basic association { 1 }u~l 5) "* (40)s~3) ~ L = 0, 2, 4, some of the low-lying SUsdg(3) irreps (2,#) in the symmetric U ~g(15) irrep {N} are (for N 1> 4),
{N} --. (2, #) = (4N, 0) q) (4N - 4, 2) ~ (4N - 6, 3) ~ (4N - 8, 4) 2 0) --.. (12)
Note that the (4N - 8, 4) irrep occurs twice (multiplicity denoted by • = 0, 1). It is to be mentioned that Akiyama (1985) gave a novel prescription for obtaining the multiplicities ct for the class of SU(3) irreps that satisfy the condition 4N = 2 + 2/~. The complete reduction of {N} ~(2,#) follows from the method of Kota (1977). The SU(3) Casimir operator and it's eigenvalues are
C2(SU~a0(3)) = 3[Q2(s)'Q2(s) + L'L] (C2(SU ng(3)))~.,~ = 22 +/~2 + 2# + 3(2 + #). (13)
The SU(3) irreps (2,#) generate rotation bands with band head quantum number K = min(2,/~), min(2,/~)-2 .... , 0 or 1. Figure 4 shows the band structure in the SU~dg(3) limit. Two important results here are that, in addition to the usual ground state (GS), beta (•), gamma (7) bands, one has odd-K bands (K" = 3 +, 1 + ) that are absent in sdlBM and two (4N- 8,4) irreps against one (2N- 8,4) irrep in sdlBM. From the structure of the SUsag(3 ) intrinsic states (Hecht 1965) shown in the inset to figure 4, it is clear that one of the (4N- 8, 4) irreps labelled (4N- 8, 4),= o is two phonon in character and the other (4N- 8,4),= 1 is hexadecupole vibrational in character and they can be distinguished by TNT as discussed ahead in § 2.3. Instead sdg Interacting boson model 425
iN+l) S~+)
Z"-'(I1,) 2 zNrz (4N-4,4)a= 0 (4N-4.4)a= I 14 I r 3 r L'I ZN -q,5,2 KY.O+Kr.2÷ K~4+ K'w,O+ KV.2 +
~Nn I (4N-2.3) 4 T 3* 4*~r- {4N,2) K'.t + K'.3 + ~.~-~'To- t- , ',,,o-, ~'-- 1--+ ~" , r-- i 4* Kit.o; i K"Z+ 2~i I {Nq}: s~-) ~:N+I 4÷-- ¢ t 3 ~ ~.N-I (4N+4.0) o*2+ Z+ I _I (4N-4,0) o: Kr =0~ * J
" 2, ~1 ;i -I {N}:(4N.O) K 1¢ =OGs.÷ L'r-O+~--~ ~:N
Figure 4. Low-lying spectrum in the SUng(3) limit for N + 1 boson system. Also shown are the GS of N-boson system and the GS band of N- 1 boson system. The intrinsic structure of the bands (SU a0(3) irreps) are given and they are defined in terms of the single boson states E, H1, Q3/2, F2 defined in the inset to the figure. Analytical expressions (Devi and Kota 1991b) that are accurate for large-N (N/> 8) for N--*N + 1 and N--*N - 1 TNT strengths S~+) and S~-) respectively are given in the figure. The two (4N-4,4) irreps for N + 1 boson system are distinguished by the = label = = 0 and 1 and they correspond to W = 0 and 1 of Akiyama (1985); for all the other irreps ~tW) = 0 and the corresponding labels are dropped. The dashed arrows indicate the TNT selection rule in the SU ag(3) limit. of the c~ label one can use (as it is not a group label) Akiyama's (1985) W label and the S operator of Akiyama (S = [Bt(04)/~(40)](°°); Bt(04) = L~(40)l-h* ~fit (40)/JVI (°4) where b t is single boson creation operator) is diagonal in the W-basis for the (~., #) irreps with 2 + 2/~ = 4N. It should be mentioned that W = 0 for the irreps with no multiplicities. A simple hamiltonian and the corresponding energy formula in the SUsdg(3 ) limit are H = c + #~ W(2N- W+ 3) + ?L(L+ 1). (14) Similarly the analytical formulas for the E2 transition strengths and the quadrupole moments of the GS band are (with T ~2 = aQZ(s)), L- 2)=o2F 2L(L- 1)(4N - L + 2)(4N + L + 1)] B(E2;(4N, O) L--*(4N,0) L (2L- 1)(2L+ 1) d a 1/I~nFL(8N+3)1 Qz((4N'O)L)=- x]-~-k (~--F3) _1" (15) 426 Y Duroa Devi and V Krishna Brahmam Kota It should be clear from (15) that in the GS band the E2 strengths grow up to L ~ 2N (against L,-, N in sdIBM) and falls to L = 4N and the band cut-off extends from L = 2N of sdIBM to L = 4N. SU~dg(5) limit: In the SUsdg(5) limit the group chain is U,no(15) = U~o(5 ) = SU dg(5) Odg(5) = Ong(3). The generators of U~dg(5) are GLo=°-4u = ,~.t:=o,2,, 2 [-(2:1+ 1)(2:2 -1- 1)]1/2(-- 1)Lo{21:2 2 L° } (bet~ b'e~)L°" (16) whereb*o=st, bt2=d~,bt,=o*andbesimilarlydefined. In(16)thesymbol{---_} stands for Racah's 6-j symbol. The operators GL°=x-4u generate SU,dg(5), GL°=l'3t~ generate Odg(5) and G~t which is proportional to L (11) generates Ong(3) group. With the basic association { 1}u{x 5j + {2}sv(s}' { [0] ~ [2] }o{5)+ L = 0, 2, 4, the GS in the SU,ng(5) limit is I {N} {2N, 0, 0, 0, 0} {2N, 0, 0, 0} [0, 0] L = 0 > ; figure 3 gives notations for irrep labels. The U,dg(5), SUdo(5) and Od0(5) Casimir operators and their eigenvalues are 4 5 Cz(U~g(5))=4 ~ GL°'GL% (C2(U~g(5)))/I1 = ~ f~(f~ + 6 - 2i) Lo=O i=l 4 C2(SUsdo(5)) = 4 Y~ GL°'GL°; LO=I a If 4 '~2 (C2(SUsdff(5))) {m} = ,=~Em,(m, + 6 - 2i) - 5~,--~z mi) ; C2(Od9(5))----8 E GL°'GL°' Lo = 1,3 (Cz(Odg(5)))['""l = ~1(zl + 3) + ~2(¢2 + 1). (17) SUng(6) limit: In the SU,49(6) limit the group chain is U~g(15) = U,dg(6) ~ SUsdg(6) SPdg(6) ~ Odg(3). The generators hLo=O-~ 5 of Usdg(6) are, h1°=°-s=u ,l.e:=~ 0,2,4 [(2f* + 1)(2f2 + 1']1/2(-1)L° {E5)2 E5;2 L;2 } (bet ~e2)uz°" (18) The operators h~°=1-5 generate SUed,(6), huz°= 1,3,5 generate SPd,(6 ) and h i which is proportional to L (11) generates Odg(3) group. Using the basic association { 1}~15)--' {12}stJ-'~{(0)(~(12)}s-'6L-0't~,~,, ''* -- 2, 4, the GS in the SU~do(6) limit is I{N} {N, N, 0, 0, 0, 0} {N,N,0,0,0} (0,0,0) L= 0); figure 3 gives the notation for the irrep labels. The U~o(6), SU~dg(6) and SPdg(6) Casimir operators and their eigen-values are 5 C2(U~ng(6))=4 ~ hr'°'hL° Lo=O sd 9 lnteractin 9 boson model 427 6 (C2(Usng(6))){r} = ~ F,(F i + 7 - 2i) i='l 5 C2(SUsag(6)) = 4 )-' hL°'h L° LO=I (C2(SUsda(6))) ~MI = M~(Mi + 7 - Mi i=l i C2(Spdg(6))= 8 ~ hL°'h z° L0 = 1,3,5 (C2(Spdg(6)))(~"x"x~> = 21(22 + 6) + 22(22 +4) + 23(23 + 2 ). (19) O dg(15) limit: In the O dg(15) limit the group chain is U d.(15 ) = O n.(15 ) = Od.(14 ) Oa,(5 ) D On,(3 ). Using the basic association { 1}u~xs~ ~ 01O,lS,- {t0] [1] }o~x,) { [0] ~) [2] }o~5)--' L = 0, 2, 4, the GS in the O d,(15 ) limit is I{N} IN] [0] [0, 0] L = 0); figure 3 gives the notation for irrep labels. The pairing operator Psdg in sdg space is expressible in terms of the Usdg(15 ) and Oag(15 ) Casimir operators. The explicit form of Psng and its eigenvalues are 1 P,. = s+ s_., s+ = Y b*.tb*.t, s_ = (S+)* ~t = 0,2,4 (P n,) INlt'3 = (1/4)[N(N + 13) - (C2(O d,(15)))/N/E'1 ] = E¼(N - tr)(N + tr + 13)]. (20) Another useful result is that [C2 (Oa0(15)) - C2 (Ong(14))] has the simple quadrupole - quadrupole + hexadecupole-hexadecupole form: C2(O,a,(15)) - C2(On,(14)) = 12-12 + I'.I'; I~2 = (s'd+ d*sO]; I~ = (s*O + 9*s')2. (21) Weak-coupling limits: In the U~d(6) ~ Ug(9) limit the U~d(6 ) group generates sdlBM symmetries Ud(5), SUa(3) and O~(6) and the properties of these groups are well documented (Iachello and Arima 1987). The Ug(9), Og(9) and Og(3) generate #-boson number n v seniority vg and angular momentum Lg respectively for g-bosons (similarly vd and L~ are defined). The ng = 0 states correspond to sdlBM and ng = 1 states generate the hexadecupole vibrational levels/bands as described ahead in § 2.4. In the Udg(14) limit, nag = nd + ng is a good quantum number, and the other groups Odg(14), Odg(5 ) and Odg(3) in this chain are already discussed. Finally, in the Ud(5)~ Usg(10) limit two chains, one with U g(10)~ Osg(10) and the other with U~(10) Ug(9) ~ Og(9), as shown in figure 3, are possible. The group O g(10) generates seniority in sg boson space and the Ud(5) chain corresponds to sdlBM Ud(5) limit. In order to understand the physical relevance of the various dynamical symmetries, the Wigner-Racah algebra of the group chains given in figure 3 has to be worked out. The algebra is available for the SU dg(3) limit (Elliott 1958a, b; Hecht 1964, 1965; Arima and Iachello 1978b), U~a(6)0)Ug(9) limit with ng = 0 (Arima and Iachello 1976, 1978b, 1979) and n o = 1 (Devi and Kota 1991 a). In addition, some preliminary attempts 428 Y Dur#a Devi and V Krishna Brahmam Kota are made for U(5) = 0(5) = 0(3) chain relevant for SUug(5) limit (Vanthournout et al 1988, 1989) and U(9)= 0(9)-~ 0(3) chain relevant for Usa(6)t~ Ug(9) limit (Yu et al 1986; Sen and Yu 1987). An alternative to Wigner-Racah algebra is to use the CS formalism. This gives information on geometric shapes and also large-N limit expressions for various observables and this topic is dealt with in the next two sub-sections. 2.2 Geometric shapes In order to exploit (and apply) the rich group structure of 0IBM, it is essential that the physical structure (geometrical relationship to the Bohr and Mottelson (1975) description of collective states in terms of shape variables) behind each of the seven dynamical symmetries of gIBM should be understood. There are basically two distinct approaches to solve this problem: (i) defining operators whose expectation values in the representations of the various dynamical symmetry groups give directly the shape parameters (Chacon et al 1987, 1989); (ii) using the so called projective CS (Iachello and Arima 1987). The latter approach is used extensively with success, in studying geometric aspects of the sdIBM and Devi and Kota (1990b) adopted the same for 0IBM. In sdgIBM N-boson projective CS IN;/32,/34, 7> is constructed using the following assumptions: (i) the surface R(O, ?p) is quadrupole + hexadecupole deformed; (ii) surface is reflection symmetric with respect to X - Y, Y- Z and Z - X planes of the intrinsic system of quadrupole part; (iii) parametrizing the surface in terms of quadrupol¢ and hexadecupole deformation parameters/32 and/34 and asymmetry angle ~ such that there is one-to-one correspondence between these variables and the surface they define (Rohozinski and Sobiczewski 1981); (iv) using a simplified version of Nazarewicz and Rozmej (1981) parametrization. The explicit form of the CS IN;/32,fl4, 7> is, IN;/32,/34, ~> = [N!(l + ~ +/3~)"]- ,/2 × {S*o+/32 [cos 7d*o + 2-,/2 sin ?(d*2 + d*_ 2)] + ~/3,, 1-(5 cos 2 ~, + 1)g*o + (15/2) '/2 sin 27(g*2 + g*__2) + (35/2)1/2 sin27(gt4 + gt_ 2)] }N[0> (22) where/32/> 0, - oo ~ f14 ~< + ~, 0 ° ~< 7 ~< 60 °. When/34 = 0, the CS in (22) reduces to the CS employed in sdIBM studies (Iachello and Arima 1987). Given the general (1 + 2)-body gIBM hamiltonian (1) geometric shapes for the corresponding system can be studied by evaluating the energy functional E(N;/32, f14, V), E(N;fl2,/34, 7) = dE=o, --=dE O, and--t3E=0. (24) a/32 a/~4 a? The equilibrium shape parameters (/32,/34,o o Y o ) corresponding to HgmM are given by equations (23, 24). The compact formulas for the CS expectation values of each of sd9 Interactin9 boson model 429 Table 1. Algebraic expressions for coherent state matrix elements of various parts of general glBM hamiltonian in terms of ('82, '84, Y). t'l {2 {3 /4 L CS Matrix Element*'* 0 0 0 0 0 1 0 0 2 2 0 (1/s)'~fl~ 0 0 4 4 0 (1/3)fl42 2 2 2 2 0 (1/5)//~ 2 2 4 4 0 (1/3xf5)'8~'8~ 4 4 4 4 0 (1/9)'8~ 0 2 0 2 2 ,822 0 2 2 2 2 -(2/7)l/2fl~cos37 0 2 2 4 2 (2/7)I/2f12f14 0 2 4 4 2 --(lO/3x/~)f12'82.,cos37 2 2 2 2 2 (2/7)fl~ 2 4 2 4 2 (2/7)fl~fl] 4 4 4 4 2 (100/6237)'8~.(16- 7cos z 37) 2 2 2 4 2 - (2/7)fls2 '84 cos 37 4 4 4 2 2 -(20/21,,/22)flz'83cos3y 2 2 4 4 2 (lO/63)(2/ll)~t2fl~'sz,(4--cos23~ ,) 2 4 2 4 3 (1/9)fl~fl2,(COS2 37 - 1) 0 4 0 4 4 f12 0 4 2 2 4 (6/x/76)fl~'84 0 4 2 4 4 -2(5/77p/2'82f12cos37 0 4 4 4 4 (1/9 2x~)'8~(62+ 100cos 237) 2 2 2 2 4 (18/35)'8~ 2 4 2 4 4 (5/231)f12flz(5+7cos237) 4 4 4 4 4 (1/27027)fl~(2974 + 1400cos 2 37) 2 2 2 4 4 -(6/7)(2/11)1/2'83'84cos37 4 4 4 2 4 - (36/77)(5/26) 1/2 f12 fla3cos 3'? 2 2 4 4 4 (1/21 7~)f122f12(62+lOOcos237) 2 4 2 4 5 0 2 4 2 4 6 (l/99)f122fl](49-gcos23y) 2 4 4 4 6 --(tO/33)flzflaacos37 4 4 4 4 6 (20/891)fl~(2 + 7cos 2 3),) 4 4 4 4 8 (70/11583)fl~(79- 16cos237) ( n~ >~s = ( N; &, &, y ln, IN; &, '8,, 7 > = N/(1 + '8~ + '8,~) (r/a)cs = (N;'sz,'84,7]FlalN;flz,f14,7) = N'8~/(1 + fl22 "4- '84)2 (no) CS = (N;'82,'84,ylnglN,'82,'84,y)~ . = N'84/(12 + '822 + '84) 2 *Given a general two-body term (b~,b*/.)L-(b/f/Q L, the CS matrix element (N, f12, fl4,)l(b~bJ2)L (b/3b/a)L IN, f12, f14, ,;) is gwen by the entries in the last column of the table multiplied by N(N-1)/ I1 +'8~ + &~)2. *The expressions given in the table acqmre a phase (- 1)L under the interchanges {l*-,dz or c'3~d 4 and they remain the same under ([l,{z)*--~(ff3,[~.). With [=0, 2 or 4 it can be seen that L= 1, 7 are not allowed. the 35 pieces in (1) are derived, and they in turn are reduced to the energy functionals E(N; f12, f14, 7). The resulting algebraic expressions are given in table 1; see Devi (1991) 430 Y Duroa Devi and V Krishna Brahmam Kota for details. In order to study geometric shapes corresponding to the seven dynamical symmetry limits of #IBM, the hamiltonian is written in each case as a linear combination of the Casimir operators (C:(G) and C2(G)) of the various groups appearing in the respective group chains (U(15)~ G ~ G'~ ..-~ 0(3)). Then the energy functionals E(N;fl2,fl4,),) are constructed and the equilibrium shape parameters are obtained via (23, 24). Strong coupling limits: The hamiltonians in the four strong coupling limits SUng(3), SUng(5), SU,du(6) and O,d~(15) are chosen to be linear combination of the quadratic Casimir operators (given in (13, 17, 19, 21)) of the various groups appearing in the respective group chains shown in figure 3. Using table 1, energy functionals in the N ~ oo limit are constructed in analytic form (Devi and Kota 1990b; Devi 1991) for the four strong coupling limits. The corresponding equilibrium solutions ~a° flo ~,o~ are given in table 2. The hamiltonians are normalized such that E(N,"f12,° f14,° ~o) is same in all the four limits. From the potential energy surfaces shown in figure 5 one sees that SUug(3) has one minimum (thus producing a deformed surface), SU,dg(5 ) limit has two minima that are displaced in energy, SUug(6) limit has three degenerate minima while the O,d,(15 ) limit is completely ~-unstable. Weak coupling limits: Determination of the ground state shapes is straightforward in the weak-coupling limits U,a(6)~Ug(9), U,d(14 ) and Ua(5)~BU~(10). In the [U~(6) = Gj ~B Ug(9) limit with G~d = Ud(.5), SUed(3), and O~(6) the ground state shapes (with a physically motivated choice for the sign of the strength of the U~(9) Casimir operator in the U~(6)~Ug(9) hamiltonian) are, just as in sdlBM (IacheUo and Arima 1987), spherical, prolate quadrupole deformed and y-unstable respectively. In the Udg(14) limit with 'a' the strength of the Udg(14) Casimir operator the energy Table 2. Equilibrium shape parameters (f12,o f14, o ~' o ) and two-nucleon transfer ratio R + for g = 0 transfer in various symmetry limits of sdgIBM. Symmetry flo flo ~,o R ÷ SUag(3) (20/7) 1/2 (8/7) 1/2 0 ° 4/N SU,d~(5) (10/7) 1/2 ( 18/7)1/2 60 ° 4IN SUng(6)*: I (25/14) 1/2 (3/14) 1/2 0 ° 2/N II (8/7) 1/2 (6/7) 1/2 60 ° 2/N III (1/14) l:z - (27/14) 1/2 60 ° 2/N 04g (15) (flo)2 + (flo)2 = 1 7 independent I/N [U d(6) = G] ~ Ug(9): G = U~(5) 0 0 ~, independent 0 SUs~(3) ,/5 o o ° 2/N O,a(6) 1 0 ~ independent 1IN U~(1)~ Ud~(14) 0 0 3, independent 0 Ua(5)~ U g(10)*: I oo 0 y independent II 0 _ ~ ~< flo ~< + oo 7 independent undefined *As shown in figure 5 SUng(6) limit has three degenerate minima with different (rio, flo, 70) values and they are labelled I, II and III respectively. Similarly Ud(5)~U (10) limit has two equilibrium solutions depending on the choice of the parameters in the symmetry defined Hamiltonian (and they are labelled I and II); see § 2.2 for details. sd 9 Interacting boson model 431 8_ E o i -io -20 ¸ 7 v & -4 2 &o -2 o# 2 SUsdg (3) SUsdg (5) E [- o -I0 -I -2O -2 0 -4 SUsdg (6) Osdg (15) Figure 5. Potential energy surfaces E (the E corresponds to E(N;fl2,fl4,~? ) given in (23)) vs//2, r4 in the four strong coupling limits SU ng(3) with 7 = 0 °, SU~ag(5) with ), = 60 °, SU~a(6) with 7 = 0 and O~g(15) which is independent of 7- functional is given by E(N;fl2,fl4,7) = aN2t$4/(1 + 32) 2 with fi2 = r2.3t_2 f12. With the choice a > 0, we have go = 0 which implies that flo2~--- 0, flO = 0 and E is independent of 7. Thus the equilibrium shape in this limit is spherical and the nucleus will be vibrational. Finally in Ud(5)t~Us~(10) limit, depending on the choice of the parameters, either flo = ~ and flo = 0 or flo = 0 and -oo ~< r4 ~< + oo and E is independent of 7. Some comments about the equilibrium shapes obtained for the seven oIBM dynamical symmetries are: 1. The weak coupling limit Udg(14) and the strong coupling limits {SU~dg(3), SUsag(5) } and {SU ag(6), Osag(15)} are similar to the sdIBM Ud(5), SU,d(3) and O d(6) limits respectively, although the SU~no(5) limit is not strictly stable deformed nor the SU~ag(6) limit completely unstable. It should be added that the U~(6) ~ Ug(9) limit is essentially (in the ground state domain) same as the sdIBM. The results clearly indicate that vibrational nuclei can be described by interpolating [U~n(6)~ Ud(5)] ~)Ug(9) and 432 Y Duroa Devi and V Krishna Brahmam Kota Uag(14 ) limits, deformed nuclei by the SUing(3), SU a0(5) and [U~(6) ~ SU~a(3)] @ U9(9) limits and the y-unstable nuclei by the O~ag(15), SU ag(6) and [U,a(6) = O~a(6)] ~ U~(9) limits respectively. 2. The equilibrium shape parameters (fl°,~,7°) are independent of the choice of parametrization. This result is verified by employing, in place of (22), Rohozinski and Sobiczewskis' (1981) parametrization (using both 3 and 5 parameter versions). 3. One important feature of the solutions given in table 2 is that we never have non-axial shapes (7 different from 0 ° or 60 °) in the dynamical symmetry limits although E(N; P2, f14, 7) contain COS237terms. The HgmMgiven in (1) and the results given in table 1 show that non-axial shapes are possible in sdoIBM unlike in sdIBM. This may be due to the fact that, unlike in sdIBM (Gilmore and Draayer 1985), the general .qIBM hamiltonian Homsl (1) cannot be expressed solely in terms of the Casimir operators of the various groups given in figure 3. One can always construct special hamiltonians (the choices follow from the expressions given in table l) and one such example is worked out by Kuyucak and Morrison (1991). With a slightly different form (compared to (3, 4)) for E2 and E4 transition operators, T k=p ~ t(k)~',~'~ tb ~t~ ?'J~,~k. k=2,4 ¢',d' = 0,2,4 t(k) = t~ek)e, /.(2)= 1, /.(4) 1 (25) t,f' , *02 ~04 ~" and a quadrupole-quadrupole plus hexadecupole-hexadecupole hamiltonian H_ h Hq_ h = ea~a + ~g~ig + x 2 T 2. T 2 + t% T 4. T 4, (26) they showed that the choice/(242)= ff(~ = 0 leads to triaxial shapes. The tff,~, are related to t tk) and they are defined for convenience, ~k) = ( g,O~'OlkO) t~k)e,. (27) With this choice the energy functional for x4 T 4" T 4 is (follows easily from table 1) of the form A(/~2,f14) + B(f12,/~4)cos237. It is seen that: (i) f~ = ?t2~ = 0, ~4) < 0 and x4/x2 > 0 gives triaxial shapes and x4/x2 < 0 gives only axial shapes; (ii) ~:4 = 0 does not give triaxial shapes; (iii) x4/x2 ~ 1 gives sdo to sg transition with vanishing quadrupole moment and non-zero hexadecupole moment. A numerical example with ea = 0, e, = 0, x2 = - 1, (?~, ?~, ~) = (0"I, 0.687, - 0.25) and (?(242~, ~#), ?444)) = (0, 0, - 0-6) is shown (Kuyucak and Morrison 1991) for Emi,(fl2, f14, 7)/N2 vs x4, for demonstrating the occurrence of triaxial shapes in sdolBM. 4. From figure 5 it follows that SU ag(5) limit admits two minima and hence in sd# model it is in principle possible, with an additional variable, to have shape phase transition (i.e. the system can tunnel from one minimum to another by changing the variable). Secondly SU,ag(6 ) limit shows that it is possible to have shape coexistence in sdglBM. From table 2 it is seen that the three degenerate minima in SU a~(6) limit correspond to quite different shapes (in fact one is prolate and the other two oblate). Thus unlike sdlBM, the sdglBM admits shape phase transition and shape coexistence. Kuyucak et al (1991a) showed, by using angular momentum projection from single intrinsic states, that these phase transitions can be driven by angular momentum. For example, with Ha_ h given in (26) and ea = 0, x4 = 0, eg = 1.5 MeV, x2 = - 40 keV, P~2) = -0-2, ?(24) = 0.69 and ~ = 0-27 (the last two numbers are SU,aa(3 ) values), it is sdg Interacting boson model 433 shown (Kuyucak et al 1991a) that the quadrupole moments of the GS band in ~92Os change sign at L= 18, (i.e.) a transition occurs from prolate shape at L= 16 to oblate shape at L = 18. In general one needs ~22)and [t4~have opposite signs and 1~24) I >> 1f~2~ I. 5. In order to visualize the geometric shapes corresponding to the CS (22), we should be able to express the shape variables (13P2,fl~,?P) that define the nuclear surface in terms of the CS variables (f12,134,)'). To this end one equates the E2 and E4 matrix elements in CS and geometric model description; the latter are given by Eisenberg and Greiner (1987). This correspondence is used in studying fl4-systematics in rare-earth and actinide nuclei (§4.1). 2.3 Two nucleon transfer Two-nucleon transfer cross-sections (and the corresponding spectroscopic factors or strengths) are one of the most valuable observables in nuclear structure as they provide deeper insights into effects due to pairing degree of freedom, deformation changes, single-particle aspects etc. (Hinds et al 1965; Maxwell et al 1966; Griffin et al 1971; Broglia et al 1974; Ragnarsson and Broglia 1976). Arima and Iachello (1977), Iachello and Arima (1987) suggested that the interacting boson model provides a natural framework for a unified description of TNT cross-sections and strengths. Analytical formulas and several comparisons with data are available in IBM literature (Iachello and Arima 1987; Betts and Mortensen 1979; Cizewski et al 1979; Burke et al 1985; Miura et al 1985). TNT in SU~ag(3) limit: Ignoring the cut-off factors (square root factors) in (7) the TNT operators for ~ = 0, 2, 4 transfer in sdglBM are, pe+=o ¢=2 __ ¢=4 =r/+oS* P+ - r/+zd* P+ = ~+4g* p¢--o = r/_Og pt--2 = r/_2 ~" p¢--4 = t/_4 ~ (28) where r/'s are free parameters. Using (28), Akiyama et al (1986) derived numerically TNT strengths for 164Er(t,p)16SEr(N= 15~N = 16) in the SU~dg(3) limit. On the other hand, Devi and Kota (1991b) derived analytical expressions using the U(15) ~ SU(3)= 0(3) Wigner-Racah algebra and the results that are accurate for large N are given in figure 4. Note that the TNT strength for E-transfer from 0~s is defined by S(±)(N;O~s~N+_ 1; Ly) =(r/+e)2l(N +_ 1;LT+ [IP[+)II_N," 0GS+ > 12fitL," (29) The t66Er(t, p)168Er data exhibits selection rules and certain other interesting features. In sdIBM K ~ = 03,04,22 etc cannot be excited because (2N, 0) ® (20) ~ (2N + 2, 0) K ~ = 0~-~ (2N- 2, 2)K~= 02 , 2?. The forbidden levels are observed experimentally. The strength to 0[, 02 (1.217 MeV), 03 (1.422 MeV)and 04 (1.833 MeV) are (100, 15, 10, 2.4) and (100, 8"4, 15.2, 0) in experiment and SUdg(3 ) limit (from figure 4) respectively. As can be seen from figure 4, the transfer to IK'= 2 22 belonging to (4N - 8,4)~= 1 is six times the transfer to IK ~ = 2 2~- and it is remarkably close to data value of 5"0 (Akiyama et al 1986). It is seen in data that the strength to IK ~ = 44 + state at 2.06 MeV which belongs to (4N - 4, 4)~ = 0 is rather weak. From figure 4 one can see that the (4N-4,4)~=o. 1 intrinsic states for (N+I) boson system are two-phonon and lg-boson type respectively. This leads to the selection rule, as can 434 Y Durga Devi and V Krishna Brahmam Kota be seen from figure 4, that the transfer to (4N - 4, 4)~ = o is forbidden• Thus, the nature of the observed (4N - 4, 4) bands can be inferred from TNT. However, with Akiyama's (Akiyama 1985) multiplicity label W the selection rule (except in large-N limit) is not exact• Another feature that provides a signature of g-bosons is that the 4~- (1.737 MeV) state (Davidson et al 1981) is strongly populated although the theoretical and experimental values do not match very well (ratio of the strength to 4~ and 4+ are 40:100 and 37:12 respectively). The excitation to 4~ cannot be one step in sdlBM while in glBM it can be as it may belong to IK ~ = 4 3~. Thus the 4~- can be interpreted as IK"=4 3~ and it is in confirmity with detailed spectroscopic calculations (Yoshinaga et al 1986)• Average fraction of f = 0 strengths to all excited states: In addition to the individual spectroscopic strengths which are usually difficult to extract out, as (Burke 1990; Garrett et al 1990; Devi 1991 and references therein) suggested, it is advantageous to deal with the ratio R+ for { = 0 transfer which is the ratio of the summed cross- sections going to all excited states to that of the GS. The ratio R+ removes the uncertainties in the normalization factors in both experimental and theoretical results and can be treated as the ratio of summed strengths; E S~7= + 0)(N,"+ 06s ~ N +_1;0;) f#GS R+- - S(i=o)(N, + . OGs~N + _+ 1;0~s) (30) In (30), ( + ) is for particle addition and (-) is for particle removal. Using the N-boson CS IN;/~2/~4~'), defined in (22), large-N limit expressions for R+ are derived (Devi and Kota 1991b). In terms of the equilibrium shape parameters (/~o,/~o, yo) given in table 2, expression for R + is 1 + (N,~2~a~• 0 0 0 Is*slN;H°H°~°) R+= 1 I(N+l;/~°/~°y°ls* [N,]~2/~47. o o o )l z _ i o)2 + ( o)2 + O(1/N2). (31) N In all the symmetry limits the tensorial nature (which is {114}(04)g=0, {114}{24}[0]f=0, {II'}{I'}(000)E=0 and {11'}[1][0][0]¢=0 for SUsdg(3), SUsdg(5), SU dg(6) and Osdg(15) respectively) of the operator p~o) clearly shows that ptO~ acting on N-boson GS excites only the (N- 1)-boson GS-and therefore R_ is zero. The CS method also gives R_ = 0 in all symmetry limits N" o o o t . o o o R_ ( ,f12f147 Is stN, fl2fl,~ ) 1 I(N --1,1~u[3,y. o go [slN,. ff2flaY o go )12 =0 in all cases• (32) Figure 6 shows the analysis of experimental (t, p) and (p, t) data for R + all across rare-earth region (neutron number N = 82-126); the (t,p) data is for N ~< 104 and (p, t) data is for N >/104. One important point here is that for hole bosons ((i.e.) bosons are hole pairs) two-particle removal (p, t) effectively becomes two-particle addition sdg Interacting boson model 435 i i i v i i i i //// i i i i i i i i 1.6 (f,p) (p,t) 0 rig(15) 0 Sm (sdglBM) 1.2 + Nd v Yb .H, (6) o Gd o W ~sdg R 0.8 ~'~ * Dy • Os I % '~ "Er -Pt I / 0.4 ~ 0 • 0 ~ t , , , i L I , J //// I I l I I I L 82 90 98 106 114 122 Neutron Number (Final Nucleus) Figure 6, Ratio of the summed TNT cross sections for all known excited 0 + states to that of the GS and as described in the text this ratio is same as the ratio R defined in (30). The data compilation is due to Burke (1990), Garrett et al (1990), Devi (1991). Experimental data for a #oven nucleus are connected to guide the eye. For neutron number N = 82-104 the (t,p) data are shown and hence R---R+ of (30). For N = 104-122 the (p, t) data are shown and here, due to particle-hole symmetry explained in the text, once again R = R+. The predicted values (#oven in table 2) of the ratio R are shown in the SUng(3) limit for (t,p) and SUng(3), SUag(6) and O ~0(15) limits for (p, t); it can be seen from table 2 that SUsd.(5) limit gives the same result as SU~o(3) limit. The large hollow circles correspond to detal~led sdglBM calculations (see §2.6) for Sm isotopes and they reproduce the peak seen in TNT data which is due to spherical-deformed shape phase transition (Devi and Kota 1992a). (t, p) and vice versa. Therefore both N ~< 104 data and N/> 104 data are for R = R +. The predictions of sdglBM symmetries shown in the figure follow from (31) and table 2. Since the data are for cross-sections and the theory predicts strengths one should not expect detailed agreements and it is more appropriate to consider the average values of the ratio over a range of nuclei so that certain special features (peaks) due to detailed structure effects get averaged out. The region with neutron number N < 88, N ~ 92 to 110 and N > 110 correspond to vibrational, rotational and ),-unstable nuclei respectively. The theoretical curves are shown only for the appropriate symmetries in each of these regions. It can be seen from figure 6 that for nuclei in the rotational region (N ,,~ 92 to 110), the average value of R is close to the SUsdg(3 ) value (note that the SU d(3) value is one-half of the SU dg(3) value). For example, in the case of the deformed Gd and Er isotopes, the (t, p) data give the value of R + ~ 0-3-0.4 which is close to SUsdg(3) value 4/N. Thus the (t,p) data for deformed nuclei is better explained with g-bosons. It should be noted that the results for SUs~g(3) and SU~dg(5) are similar and therefore the TNT data cannot distinguish between these two limits. The (p, t) data for the ),-unstable Pt, Os isotopes give the value of R + (because of hole bosons) to be ,,, 0"08 which is close to the O~d(6) or O dg(15) value 1/N. The SUng(3) description in this region, as expected, is poor and that of SU dg(6) is not very much different from the O,d(6) and O~dg(15) descriptions, validating the conclusions of § 2.2. An important feature of figure 6 is that the R + data shows peaks at neutron numbers 90, 98 and 108. Detailed numerical calculations in sdg space, for Sm isotopes, 436 Y Durga Devi and V Krishna Brahmam Kota employing a hamiltonian that interpolates various dynamical symmetries of #IBM is shown to reproduce the peak at N = 90 (which is most prominent). The origin of the peak is that Sm isotopes exhibit shape phase transition; see Devi and Kota (1991b, 1992a) for details. Thus sdglBM dynamical symmetries describe the gross structure of R + TNT data and details (peaks) are explained by spectroscopic calculations (§ 2.6) in sdg space. Finally it should be mentioned that R_ data in rare-earths are also available with features similar to those shown in figure 6 and sdglBM describes them in a similar fashion as above (Devi 1991). 2.4 Hexadecupole vibrational bands The sdglBM provides a natural frame work to understand and predict about hexadecupole (E4) vibrational states (or the collective bands built upon them) as these states (or bands) in sdglBM correspond to g (d = 4)-boson excitations. It is possible to identify low-lying bands built upon E4 vibrations by coupling a single #-boson to the core described by s(# = 0) and d(d = 2) bosons. Clear band structures emerge out when the core is described by one of the three limiting symmetries Ud(5), SU,d(3 ) or O,d(6 ) of IBM. Here below the SUa(3 ) x lg limit is described in some detail and the other two limits Ud(5) x lg and O,d(6 ) x lg are described briefly. Complete details are worked out by Devi and Kota (1991a). SUsd(3) x lg limit: The SUsd(3 ) x lg limit generates rotational bands built on hexadecupole vibrations. The hamiltonian in the SU,d(3 ) x lg limit is FQ 2 "02 - A: [(dt~)*'(gtd)4]: + k'L'L, (33) where Q~ = v/~ {(std+ dis') 2- x/~/E(dtd) 2 } is the SUed(3 ) quadrupole operator, Q2 __ (gt 9)2, :: denotes normal ordering and L (11) is the angular momentum generator. It can be shown that (Arima and Iachello 1978b; Scholten 1980; Bijker and Kota 1988) as N ~ oo and F, A >> K, the W(K, L) states given below become the eigenstates of the Hamiltonian in (33), U~(K, L) = IN;Nsd = N - 1, (2N - 2,0); g; KL> ~ a ,=,v., 1)L /2R+l/ 2 LK> = ~ (- X/2L+lX/(I+dro) E(N; (2N - 2, 0); g; KL) = - x {(2N 2 - N - 1)- ~(L(L + 1)+ 20 - 2K 2)} F(N- 1) . 2 ,f A(3K 2 - 20) --,~= (3~ - 20) 1 -t 3~/154 + eg + k' L(L+ 1). (35) sdg Interacting boson model 437 Thus one can have rotational bands built on hexadecupole vibrations. Their structure is given by (34) and their energies by (35). In addition to the lg-boson bands, there are the usual K~= 0 + GS band described by the SUsd(3 ) irrep (2N,0) and the fl, ;: bands generated by the irrep (2N - 4, 2) with K" -- 0 + and 2 + respectively. In figure 7 the energies of the W(K, L = K) levels are shown as a function of A to illustrate the important point that with the variation in A, relative positions of lg-boson bands can be altered. In the absence of the A term (A--0) the bands will be arranged as K" = 4 +, 3 +, 2 +, 1 + and 0 + or K ~ -- 0 +, 1 +, 2 +, 3 + and 4 + depending upon the sign of F. However as can be seen from the figure with A <~ - 0-15 MeV (and x, k' and F taking the vlaues mentioned in the figure caption) one can bring down the K ~ = 3 + band and the ordering of the bands for A -- - 0-25 MeV is K ~ -- 3 +, 2 +, 4 +, 1 + and 0 +. Thus it is seen that the exchange term can explain some of the low-lying K ~ = 3 + bands observed in some rare-earth nuclei. In the SUsd~(3) limit of gIBM (Akiyama E (MeV) 3 ,~=o K=I ~.~"~~._.___ ..... _ K =2 (a) -I K=4 xx x I I ~ I I I I 1 I I t I I i I 1 1 I I -0.25 0.00 0.25 0.5 A(MeV) --'4+ ,o* --':,o* --Jo* --s ÷ --~÷ . • " __,. : : :=0 __ 9 ÷ __ a ÷ -- 3+ = 2 12+ __s + __8 ÷ : K ~r=3+ > (b) 4) I0+ 4+ 5+- Klr= 4+ v ILl 2 + ~ ~ I CK'=¢ o~~* :=2~ ~ su(3)¥ x Ig ' o÷ KY: 0+ Figure 7. (a) Energies of the states IN; (2N - 2, 0); g; KL = K) relative to the GS as a function of A and calculated using (35). The total number of bosons N is 12, the ratio F/r = 5, ~= 15keV and k' = 7-7keV. (b) A typical spectrum in the SU~(3)x lg limit for N= 12. Parameters in the hamiltonian (33) are K = 15 keV, F = 75 keV, A = I00 keV, eg = 1.6 MeV and k' = 7-7 keV. 438 Y Durga Devi and V Krishna Brahmam Kota et al 1987) 3 + and 1 ÷ bands come very close whereas with A = -0"25 MeV they are separated by about 1 MeV. Figure 8 shows the typical spectrum in SUed(3) x lg limit. The E2 and E4 properties of lg-boson bands IN; (2N- 2, 0); g; KL) are studied in detail by Devi and Kota (1991a). For illustration and the analysis presented in §4.5, the E2 properties are discussed below. In the SUed(3) x lg limit it is convenient (and appropriate) to take the T le2~ operator to be of the form, T(f2' = ~(Q,u)~ + e~(d* ~ + g* d)~. (36) Analytical expressions for the reduced matrix elements of T e2 in (36) for GS--* GS, lg--.lg, GS~lg, V--.lg are (note that only the Q~ terms in (36) contribute to GS-*GS and lg~lg transitions and only the (d*O+g*d) 2 term contributes to GS ~ lg and 7 ~ lg transitions); (N;(2N, O)LIIQ2,dIIN;(RN,O)L ') = x//2N[(2g + 1)(2L' + 1)] 1/2 0 (37) (N; (2N - 2, 0), g; gz II Q~ II N; (2N - 2, 0); g; g'z') =x/2(N-1)(-1)L+r[(2L+I)(2L'+I)]I/2( L'2 L )6 K O - K rK" (38) (N; (2N, 0) L II (dr ~)2 II N; (2N - 2, 0); g; K'L') "'L'[2ON(2L+I)(2L'+I)]'/2(L'L 2 )( 4 2 2 ) (39) =(-"L K'0-K' '0-K' (N; (2N - 4, 2)KL' [I (d~ ~)2 II N; (2N - 2, 0); O; KL) F5(2L+I)(2L'+l)]l/2{(_I)L ( 2 42) = a" [20 ~-x0-~ + g~) J -(K +K) K X ( 2 L' L ) + (-- 1) L+L' ( 2 4 2 )( 2 L'?K)} ; K+K -K -K K-KK -K K-KK ao=-3-1/2; a2=l; K=0or2 (40) In (37)-(40), ( ) is the standard Wigner 3 -j symbol. From (40)it is to be noted \ / that the W(K, L) --. ),-band strength is independent of N. Thus in general q?(KL)--. • '(K' L')>> ~(KL)~GS >> W(KL)--.?. There can be some changes in this trend due to finite N effects and K selection rules. For example it follows from (39) that ~(K ~ = 3 +, 4 + L)--.GS E2 transitions are forbidden. Finally, it should be mentioned that the B(E2)'s can be calculated using B(E2; L,--. Lf) = (2L, + 1)- z I(Lr II T E2 II L~)12. (41) Equations (37)-(41) are used in the analysis of K"= 4~ band in a56Gd and K ~ = 03 , 3~- bands in 17aHf (§4.5 gives details). Ud(5) x 10 and 0~(6)x 10 limits: In the Ua(5)x lg limit, with suitably chosen Hamiltonians, one has N, N' bands with lg boson excitations in addition to the sdo Interactin9 boson model 439 ground Y, X, Z... bands (Arima and Iachello 1976); Y = band: IN; N,d = N; nd, vd =nd, Ld = 2na). N-band: IN;/V~n = N - 1; nd, vd = nn, Ld = 2nd; 9; L = 2nj + 4); na=0, 1,2,3 ..... N- 1 N'-band: IN;Nd = N - 1;na, vd = nd, Ld = 2nn;9; L= 2nd + 3); na= 1,2,3 ..... N- 1. (42) The energies, E2 and E4 properties of N, N' bands are studied in detail by Devi and Kota (1991a) and good examples for the Ud(5) x lg limit come from nuclei in Pd-Cd region. The 43 levels in Cd isotopes which are essentially one phonon hexadecupole vibrational in nature belong to N-band (42) as discussed in § 4.3. In the Osd(6) x lg limit, once again with a suitable Hamiltonian, one has X, ./if' bands in addition to the ground (~¢) and other bands (Arima and Iachello 1979); ~¢: IN; N~ = N, tr = N, ~, L~ = 2~)(~ = 0, 1, 2 ..... N; tr and ~ are 0(6) and 0(5) quantum numbers in the 0(6) limit), ~,V'-band: IN; N~d = N - 1, tr = N - 1, z, Ld = 2z; g; L = 2z + 4); z = 0,1, 2 ..... N - 1 JV"-band: IN;N'sd=N-I,tr=N-I,z, Ld=2z;g;L=2z+3); z= 1,2,3 ..... N-1. (43) The energies, E2 and E4 properties of .h/', Jff' bands are studied in detail by Devi and Kota (1991a) and examples for them are expected to come from Pt-Os and Xe-Ba isotopes although a good example is yet to be found. 2.5 sd# Hamiltonians The analytical results given in §§2.1-2.4, though they give good insight into the sdg model, have limited applicability. In order to proceed further it is important to obtain sd# hamiltonians (i.e. determine the 3 + 32 parameters in (1)) and then use them in numerical calculations that are required for a majority of (transitional) nuclei. Broadly speaking, two approaches to this rather complicated problem are available: (i) phenomenological; (ii) microscopic (based on shell model and its relatives). The symmetry defined hamiltonian Hsv M of Devi et al (1989), Boson surface delta interaction HBsol of Chen et al (1986), hamiltonian Hco M based on the commutator method given by Kuyucak et al (1991b, c) belong to the first class while the OAI mapped and IBM-2 to IBM-1 projected hamiltonian HoAvproj of Devi and Kota (1992b), Dyson-boson mapped and IBM-2 to IBM-1 projected hamiltonain HDvs.proj of Navratil and Dobes (1991) and single-j shell seniority mapped hamiltonian HoA1.full of Yoshinaga (1989a) belong to the second class. These various hamiltonians are described below. Hamiltonian HsrM: Interpolating the hamiltonians defined by the Casimir operators (linear and quadratic) of the various symmetry groups of #IBM, a schematic interaction Hsv M with a few parameters (2 + 6 instead of 2 + 32; the s-boson energy e~ can be removed for calculating excitation energies and transition strengths) can be 440 Y Durga Devi and V Krishna Brahmam Kota written down, HSYM = t;dfla + ~0f19+ gl (H(SUsdg(3))) + ct2(H(SU~dg(5))) + 0t3(H(SUsag(6))) + g4(n(osag(15)) ) + 0q(H(O~(6))) + ~trL'L; H(SUsno(3)) = - C2(SUsdo(3)) + ¼(L'L) = - 3Q2(s)'Q2(s); H(SU,ag(5)) = _ C2(SU as(5)) + ½C2(Oag(5)) = - 4[G 2.a z + G a. G4]; H(SU dg(6)) = - C2(SU d~(6)) + ½C2(Spdo(6)) = - 4[h 2.h 2 + h4"h4]; H(Osng(15) ) = C2(Osdg(15) ) _ C2(Odg(14) ) = 1-12.12 +/4.i4]; n(o~d(6)) = 12.12. (44) One important feature of Hsv Mis, as can be seen from (44), that it is indeed a mixture of quadrupole-quadrupole and hexadecupole-hexadecupole forces. Based on the results of §§ 2.2, 2.3 it is easily seen that the free parameters in Hsv M.(44) are further reduced depending upon the nucleus under study. For example, for vibrational nuclei (Pd-Cd region) it is appropriate to choose at2 = ~3 = ~s = 0, for rotational nuclei (for example, tSrsm, 16SEr) 0t3 = 0ts = 0 and for y-unstable nuclei (Pt-Os region) 0~2 = 0. The Hsv M is employed with success to describe the properties (spectra, E2, E4 .... matrix elements) of { 2°Ne, 24Mg, 32S, arAr} (Devi et al 1989; Devi 1991), {Sm isotopes} (Devi and Kota 1992a) and {~92Os, 194'196'198pt} (Devi and Kota 1991c, 1992c). Hamiltonian Hasp1: The BSDI interaction (Chert et al 1986) is constructed out of SDI, which is well known in nuclear structure (see for example Brussaard and Glaudemans (1977)), vSOl(l, 2) = -- 47th:f(r(1) -- r(2))f(r(1) -- Ro) (45) where r(1), r(2) are the position vectors of the interacting particles, R o is the nuclear radius and x is the strength parameter of the delta force. As SDI assumes the interaction to be independent of the radial part of the orbits in which the particles move, the radial part of the interaction yields constant contribution C(Ro) to the matrix elements of SDI. Putting xC(Ro) = 1, the symmetrized matrix elements for boson surface delta interaction (BSDI), using multipole expansion, are I \ Symm (BSDI)symm =4n(fo(bt~ yk(r(1))" yk(r(2))/d'd)L =[(2{, + 1)(2Eb + 1)(2ec + 1)(2E, + 1)]'2 k (1 + 6t,¢,)(1 + 6tot, ) / x 2(2L + 1)- ~(do0¢~01L0) ({c0(n0L L0). (46) The above interaction can be generalized in two ways (Devi and Kota 1990a), either by making the strength parameter to be L dependent (in (46) we have unit strength), (i.e.) HBsDI~c,,1 = ZLCz(BSDI) symmor by replacing Y~k yk. yk in (46) which has uniform sum w.r.t, k, with k-dependent summation, (i.e.) HRst~l~ak}=4rc(/,/b[EkA~ yk. yk[{,cg n)symm L and it is easy to write down its matrix elements. It should be mentioned that Chen et al employed HSSDI c,I given above in studying the spectra of {52Cr, 94Mo, 96pd} (Chen et al 1986) and { 76 •38 ' 80 •82 Kr, 76 Se} (Chen et al 1988). sdo lnteractin9 boson model 441 Hamiltonian HcoM: Instead of using HsyM, where quadrupole-quadrupole and hexadecupole-hexadecupole forces of specific types (defined by symmetries) are linearly interpolated, one can first interpolate the quadrupole and hexadecupole operators (using the general forms given in (3,4) or (25)) and then construct a quadrupole-quadrupole plus hexadecupole-hexadecupole force Hq_ h defined in (26). The H has ten free parameters With x 4 = 0 and ed, e_ ~- 0 and a suitable choice 2 q-h . A " Y of T (say T z =QZ(s) of (10)), Hq_ h describes the GS (the intrinsic state being (XooSt0 + x20dt0 + x40gto)NI0)), fl and 7-bands in quadrupole deformed nuclei. Insisting that the GS, fl, 7 bands be largely unaltered by the addition of T 4. T 4 interaction, a relationship between t t',:'~2) and t :,:'t4~ of T 2 and T 4 operators is derived by Kuyucak et al (1991b). Here the 1/N method described in §2.7 ahead is used which gives the eigen-mode condition E ~)Xeo=AkXjo; k=2,4 and j=0,2,4. (47) ~' = 0,2,4 In (47), 2k are eigenvalues and ~k) are defined in (27). It is seen from (47) that x simultaneously diagonalizes ~k) for all k and therefore the ~k) matrices commute for ~J: (4) all k. Applying (47), the relationships between t!2),~ and tij are ~-(444)= -(2) -(2) - i -(2) 2 __ ~'(2)~'(2)~ t24 Jr (t24) ((t44) -- 1 .22~44,. (48) With the condition (48), Hq_ h is defined to be HcoM, which has seven free parameters and the important property that the low-lying bands generated by Hcou are essentially quadrupole in nature, and hexadecupole effects are in the high-lying bands (K = = 3 +, 4 +, 1 + etc). The Hco u defined by (26, 48) is employed in studying K ~ = 3 +, 4 + band systematics in rare-earth nuclei (Kuyucak et al 1991c) and y-unstable nuclei 194'196'198pt (Kuyucak et al 1991b). Hamiltonian Hoat.proj" In order to construct hamiltonians with microscopic (shell model) basis one has to start with proton-neutron (p- n) degrees of freedom. The resulting p- n sdglBM is discussed in some detail in § 3.1. A simple and yet useful microscopic model is to assume that the single boson energies derive from identical particle (pp and nn) interaction and SPE in fermion space and that the TBME for bosons derive from p- n (or 7zv) interaction (Devi 1991; Navratil and Dobes 1991). With this, assuming quadrupole-quadrupole plus hexadecupole-hexadecupole form for ~ - v (or p - n) force, the p - n sdglBM hami!tonian is HvIBM.2 = ~'~' (F,dplldp Jr 8gp~Op) Jr K(2)f)2"O2.~z. ~v + Kt4)r)4"t~4~.~. z~, (49) p=/t,V where ~np and ~ are d and g boson number operators for p = n or v. In (49), edp, , K (z) and K(~fare free parameters. Using the OAI correspondence (Otsuka et al gp nv n 1978), with vp the seniority quantum number, i(jp)2 N,,, vo = 0, Jo = 0)*--, Insw = N o, Lp = 0) I(jo)zu,',vp = 2,Jp = 2>~ln~;o --- N o - l,na: p = 1 L o = 2> [(jp)2u:, vo = 2, Jo = 4> ~]nw = N o - 1, now = 1 Lp = 4> 442 Y Duroa Devi and V Krishna Brahmam Kota wherejp = (2tp - 1)/2 with 2f~,(2f~) being the shell degeneracy for protons (neutrons), and equating the matrix elements of multipole operators in fermion (rao Y~(Op, 49p)) and boson (QuX:p) spaces one obtains the effective charges e~.;p that define Q~;p and they are eta' ,,,~, (2(tip_No) ~1/2 fo;p = ~0e;0 = \f~p(f~p _ 1)(22 + 1)/ e{~l = e{~} __T_(f~p-2N_p'][4(2t~+l)(2F+l)]l/2{f f' 2} ee';p f'f;P- \ f~p-2 J (22+1) pjpjp # # t" and 2 = 2, 4. (50) The (-) sign for ete~,;, in (50) is for particle bosons (fermion number Ny <~ f~,; N, = N¢/2) and the (+) sign is for hole bosons (fermion number Nf/> tip; N, = (2tp - Nf)/2). The factor (jp II r ~ Y~(Op, 4Jp)IlJp) that appears in the mapping is not shown in (50), P and is absorbed in the free parameters K {a} Note that (j~,j,) takes values (31/2, 43/2) and (43/2, 57/2) for rare-earths and actinides, respectively. Now carrying out IBM-2 to IBM-1 projection (Frank and Lipas 1990) by assuming that the low-lying levels belong to F-spin (Lipas et al 1990) F = Fm~, = (N~ + N,)/2 and using the simple result that 2=2,4. (51) In (51) :: denotes normal ordering. Assuming ed and % to be free parameters (instead of deriving them from pp and nn interactions) the usefulness of HoA~.p,ojis demonstrated (Devi and Kota 1992b) with 15°Nd as example and the results are discussed in §§ 2.6 and 4.4. Harailtonian Hoys.p,o/ A more realistic microscopic derivation of sdff hamiltonians where the single boson energies are obtained via mappings and also the multi-j shell case is addressed (both of these are missing in HoAi.p,oj of (51)) is due to Navrali and Dobes (1991). They adopt Dyson boson mapping where bi-fermion operators a atwp}t,,t a~a# and a~ a Bare mapped onto boson operators B~p, B p via a non-hermitian mapping, atat=B~p--g'Bt B t B* = a~ap=EB~rB#r. (52) 7,~ The derivation of HDvs.proj involves: (i) rewriting (52) in angular momentum coupled form; (ii) transforming the 1 + 2 body fermion hamiltonian H I in terms of at's and a's into boson hamiltonian in terms of B t's and B's using (52) to give the hamiltonian sdg Interacting boson model 443 Hays; (iii) as Dyson mapping is non-hermitian Hay s will be a triangular matrix (i.e. Hoe s = Ha + W where Ha is the diagonal part and the eigen values of Hoe s are same as that of Ha). Using a similarity transformation, HDes can be transformed to Ha basis. Taking H I to be the pairing operator, Ha will have good seniority and in this case the similarity (seniority) transformation is worked out by (Geyer 1986; see also De Kock and Geyer 1988). Applying this seniority transformation to Hays the hamiltonian Hsen is obtained; (iv) hermitizing the Hen by using a prescription given by Navratil and Dobes (1990); (v) applying (iii) and (iv) to transition operators; (vi) identifying s, d and g bosons as the bosons B~= o, B~= 2 and B~= 4 respectively; (vii) as in mulit-j shell case there are many J = 0, 2, 4 bosons, the s boson is defined to be the collective Bto state which is obtained via BCS prescription or some other variational procedure and similarly d and g bosons by using TDA or its relatives; (viii) transforming Hay s into collective and non-collective boson operators and throwing away all non-collective parts and also the cross terms, retaining the lowest order terms in collective s, d and g bosons; (ix) using (i)-(viii) above, p- n sdglBM hamiltonian is obtained and just as in the HoA~.proi,pp and nn interactions are used to obtain single boson energies and the p-n interaction gives TBME (~- v interaction); (x) finally carrying out IBM-2 to IBM-1 projection to give Hovs.vror Employing the hamiltonian (49) Navratil and Dobes (1991) determined the single boson energies edp, egp using fermion single particle energies and surface delta interaction together with (i)-(x) above. Similarly effective charges ett~7.2'4~ that define Q2 and Q~ are determined using (i)-(x). The edp, e p and etch7;2'4, are tabulated for Z = 50-82 and N = 82-126 shells. With these parameters HDvs.projis constructed and applied to vibrational 14aSm, nearly rotational 15°Nd and 7-unstable 196pt with remarkable success. The details are given in Navratil and Dobes (1991) and some of the results are discussed in §§ 2.6 and 4.4. To the authors' knowledge these are the best microscopic sdglBM calculations available so far. Hamiltonian HoA~.f,u: The methods employed in HoA~.p~ojand navs.proj do not deal with the full fermion hamiltonian (the boson TBME coming from pp and nn interactions are dropped). Yoshinaga (1989a) dealt with this ,problem employing single-j shell OAI mapping and used the resulting hamiltonian to carry-out some interesting model studies in sdg space (Yoshinaga 1989a, b, 1991). Starting with the SDG pair states in single-j shell, S t=At(0), D t=PA t(2), Gt=PAt(4) At(J) =, 1//-~(at. a t. )J, (53) V ' J J where P is projection operator (Otsuka et al 1978) onto states of maximum seniority; note that (53) is a different form of OAI correspondence given below equation (49). In order to derive TBME one needs to construct 4-fermion states in terms of S, D and G pairs. However, the 4-fermion states I(S)N- 2 (D)Z), I(S)N- 2 DG) and I(S)s- 2 (G)2 will not be in general orthogonal to each other. They are orthonormalized using a standard procedure and the using OAI prescription (i.e. establishing correspondence between SDG and sdg states and equating the matrix elements in fermion space with the corresponding ones in boson space), the mapped interaction HoA~.ful~is obtained; Yoshinaga (1989a) gives expressions for the 3 + 32 matrix elements in (1) in terms of single-j shell fermion matrix elements. For example, the mapping for the pairing 444 Y Durga Devi and V Krishna Brahmam Kota operator P = - A *~°)A~°) is trivial and the corresponding boson hamiltonian is, H Ipairingl -- s*~ + - {(1/2)s*s*gg + s*dt'dg + s*gt'~tg} • o t-r . = I (2j 4+ fi 1 (54) Starting with a model hamiltonian Hs=-x(AtC°)At°I)-(1-x):Q2.Q2: and Q2=(a~)2, Yoshinaga (1989a) showed that in vibrational case (x=0.7), for (j = 23/2) 6 system, the sdg H gives a more complete description of the exact spectrum (several of the levels lie outside the sd space). In the transition class (x = 0.3) the sdg description (moment of inertia, binding energy etc .... ) is quite good. In the rotational case (x = 0) with sufficiently large number of bosons sdg model generates good rotational bands. In a N = 5 boson (10 fermion) calculation with n o ~< 0, 1,2,3,4,5 Yoshinaga showed that with n o ~< 3 one obtains results close to exact. Another important conclusion of Yoshinaga's (1989b, 1991) work is that SDG pair truncation of fermion space gives results essentially same as exact and hence sdglBM is close to reality (unlike sdlBM). The above method can be employed in p, n spaces separately and once again as in HoaI_proj and Hovs.p~oj one can carry out IBM-2 to IBM-1 projection to get more realistic sd# hamiltonians. Some attempts are also made to obtain HoA~.r~li in multi-j shell case (Yoshinaga and Brink 1990). 2.6 Spectroscopy in sdg space Spherical basis: It should be clear at the outset that for most of the nuclei (being transitional in nature), #-boson mixing calculations are essential: a mixing of (sd)N, (sd)N- 1(#)1, (sd)N-2(g)2 ..... (st0 ~-", (g)~, spaces is to be carried out. The truncation of the sdg space with nglnax = 1 is adequate for vibrational nuclei and for deformed and y-unstable nuclei n~< 2- 3 is adequate. For example, Pannert et al (1985), using a microscopic theory based on HFB intrinsic states showed that even for a well deformed nucleus 154Sm n m~g = 2. With truncations based on ngmax , an appropriate basis is the spherical basis In,; nd, va, ~td, La; ng, og, %, La; L) (55) where ~'s are multiplicity labels which are required when a certain L value occurs more than once for a given v. A further truncation of the space defined by (55), besides the, n ="~ restriction, is to impose that n,/> n rain, , with. n,-min ~ 2. This restriction is drawn from the knowledge that (/t,)u,~5~s ~= N, (~lx)su,,~3)Gs =(N(2N+ 1)/3(2N- 1)) and (~)~s~6~ = (N(N + 3)/2(N + 1)); typically N ~ 5 - 10. For example, with n~/> 2 and ng ~< 2 the matrix dimensions for (L = 0,2, 3,4) are (29,73,63,95), (41,114, 105, 153), (58, 167, 160,231) and (77,231,231,328), respectively, for 14SSm, lS°Sm, lSaSm and 1S4Sm" Computer codes, for constructing and diagonalizing hamiltonian matrices for general sdg hamiltonian in (1) in the basis (55) and calculating matrix dements of various operators defined in (2)-(7) using the resulting wave functions, are developed and documented by Devi and Kota (1990a). Spectroscopic calculations, for 0~, 2~ and 4~- levels in 146-xSSSm which exhibit spherical-deformed shape phase transition are carried out in sdg boson space with the restriction n,~>2 and na~<2. Observed excitation energies Ex, the ratio R=Ex(4~)/Ex(2~)," two neutron separation energies $2~, quadrupole moments, sd# Interacting boson model 445 transition moments Q(01 --, 2 ~- ) = [(167t/5) B(E2; 0~s- 2~ )] 1/2, magnetic 0-factors, E4 transition strength B(E4T)= B(E4; vGSn + --,~t+),-- 1 isomer (,~(r2)) and isotope (A(r2)) shifts (in (5, 6) ~2 instead of r 2 is used) and two nucleon transfer strengths for ¢ = 0 transfer are reproduced with a simple two parameter hamiltonian H = F {~and + eong + xQ2(s).Q2(s)}. Here (F,x) are free parameters, and ed and eg are determined from microscopic considerations. This hamiltonian is a simpler form of Hsv M defined in (44) and H t_ h defined in (26). The results of the calculations are shown in figure 8 and details regarding the parameters in the hamiltonian and various other operators are given in Devi and Kota (1992a). The agreements shown in figure 8 demonstrate that matrix diagonalization in a truncated sdg spherical basis is well suited for describing the structure of complex nuclei. However, the above two-parameter hamiltonian does not reproduce the fl and 7 band energies and other related properties (the earlier HB calculations of Otsuka and Sugita (1988) suffer from the same problem (Yoshinaga 1991)). This calls for more elaborate hamiltonian and with this reliable predictions for E4 distributions (see § 4.4 ahead) and other detailed properties can be made. To this end full Hsy M defined in (44) is employed and 148-154Sm are studied I I ] I I I 1 (o) ,l-I',p) (i)-I ::L +-- ~g I 4 + Cd U" ILlX A 0 ,a.. °Jg~ __~ I I I I i II o~E v -(P~)., (k)-T Ol "--~'01 'O 15 x e41. .~ II L t t ~ f I f (,O 0 I L L I I 1 I! N E o5~ ~r~ (h) I O, --,,'0 2 (£) v ~J v<~ 0.2~ ° T-"I I I " ~ t I I J 14.6 158 146 158 146 158 A A A Figure 8. Comparison of properties of 0~, 2~ and 4~ levels between calculated (full-line; Devi and Kota 1992a) and experimental (points) for 146Sm to 158Sm with boson number N varying from 7 to 13. (a) Excitation energies (Ex) of 2~ and 4~ levels (Lederer and Shirley 1978). (b) Ratio R = Ex(4~ )/Ex(2~ ). (c) Two nucleon separation enegies (Wapastra and' Bos 1977). (d) Transition moments of 0~--,2~ (closed circles; Raman et al 1987) and static moments of 2+ states (open circles; Raman et a11987; Powers et al 1979). (e) 0-factors {Kugel et al 1972). (f) B(E4;0~s---,4~) values (Cooper et al 1976). (g) Isomer shifts (Kalvius and Shenoy 1974). (h) Isotope shifts (Lee and Boehm 1971). (i) 0~--*0~ (t,p) cross sections (Bjerregaard et al 1966). (j) 07 --,02 (t,p) cross sections (Bjerregaard et al 19C6). (k)0~ --,0~ (p, t) cross sections (Debenham and Hintz 1972). (i) 0~ ---,0~ (p, t) cross sections (Debenham and Hintz 1972). 446 Y Durga Devi and V Krishna Brahmam Kota 4+~'-.__ 4+ 2 54"N• __6 + ~--5 + 4 + ..... 4 4. .-- 3 + / s4".... s+ 3+~, ' % + "--4 2+ .~2 + 4+ 3--- + --'-.__ 4+ "'--3 2+ Expt Theory A 64...... 6+ 2+ ..... 2+ > ",__2 + quosi- y > 0~ 2+--, Expt Theory ~E 4+ ..... 4+o +_, ~ I "--2 + qu0si-y 4+ ..~4 + , LU I.IJ -- 0 ..... 0 + Expt Theory Expt Theory 2+ .... 2+ quasi-/B 2, ..lZ+ q,,si-0 148Sm 15%m 04"__.~ 0 + O+..... 0+ Expt Theory rxp! Theory quosl -GS quosi-GS __64. ~6÷4+ _.__,÷ __ 5+ ÷ 3 + /.~- 6 + + 4 + '4 ..... Expt Theory + --z~. > z+~ > 2+~-" 0 T (1) 4+ ..... at "..__ 2+ a~ 0 .... Dtpt Theory :E Expt Theory > I B bJ Z --'----2 T 6+..... 6+0+ ..... o+ ,,~ Expt Theory 4÷ ..... 4+ 4+ .....4 + 154Sm z* ..... 2* 152Sin o + ..... o* o 0 ..... 0 Expl Theory E xpt Theory GS GS Figure 9. Comparison between calculated (theory) and experimental (expt) low-lyingenergy levels in GS,/~ and 7 bands in l*SSm, lS°Sm, lS2Sm and 154Sm. As the nuclei 14s'lS°Sm are not deformed GS, # and 7 bands in these nuclei are labelled quasi-GS, quasi-/~ and quasi-y bands respectively. The theoretical calculations are due to Devi and Kota (1992a) and the experimental data are from Scholten et al (1978), Lederer and Shirley (1978), Scholten (1980) and NDS (1990). The hamiltonian employed in these calculations is defined in (44) and the values of the parameters (in keV) (ed,%, ~q, ~2, %, ~t4,%, %) are (900, 1950, 3.87, 2, 0, 6.17, 0, 8'55), (620, 1450, 1'95, 3.4, 0, 6"52, 0, 2'41), (700, 1200, 3.44, 16.68, 0, 30-24, 0, 11'95) and (700, 1200, 4-47, 8'46, 0, 31"17, 0, 9"75) for 1'8Sm, lS°Sm, lS2Sm and 'S4Sm respectively. Note that in these calculations the strength of H(SU~g(6)) and H(O~(6)) defined in (44) are put to zero as the nuclei under consideration exhibit vibrational-rotational phase transition. As the H(G) in (44) are in multipole-multipole form they contribute to d and g boson energies. Adding this contribution the (sj, eg) values above change to (900, 2110), (580, 1460), (480, 1180) and (420, 1100) keV for 14aSm, l~°Sm, 152Sm and 15'Sm respectively. These e's show gradual decrease, as expected, from vibrational 14SSm to rotational t 54Sm. by Devi and Kota (1992a) and the spectra for 148,15o, 152,154Sm are shown in figure 9. These calculations keep intact all the results shown in figure 8, Similarly the spectra obtained for 15°Nd using HoA]_proj with the restriction ns >~ 2 and ng ~< 2 (Devi and Kota 1992b) and HDvs.orojwith ng ~< 3 are shown in figure 10. The calculations provide a consistent description of E2 and E4 properties and the results for B(E2) values are sdg Interacting boson model 447 2.0 r-Z :.s Z --"--:--g g_ _ g --2 + "~,'.,'-- 3 + 4 + __ ~; 3 + -- -- 2 + >1~ 1.0 __ 2 +- -''' ...4 0 + 2 + ; ,: 6+ ...... 6 + i -- O+ _. 0.5 4+ .... 4+ 15ONd 2+ 2+ t 0.0 0+ 0+ sdg exp sdg sdg exp sdg sdg exp sdg OAf DYS OAI. DY$ OAf DYS prol proj prol pro| proj prol Figure 1O. Experimentaland calculated energylevels for ~S°Nd. Experimentaldata is from NDS (1986). The results calculated using HoAl_proj (Devi and Kota 1992b) and HDYS.proj (Navratil and Dobes 1991) in sd9 space with (n~a'= 2) and (n~"x=3) are labelled as sdg--OAI--proj and sdg--DYS--proj respectively.Note that the boson number N = 9 for ~S°Nd. Calculation details are given in the respective references.It is important to note that the simple HoAl.proj gives results that are in closer agreement to data compared to the results obtained with more microscopic HDYS.projalthough the number of free parameters are same in both the calculations. shown in table 3 while E4 properties are discussed in §4.4. Finally, in figure 11 spectrum for the 7-unstable nucleus z96pt calculated using Hsv M (44) with ns i> 2 and n o~< 2 (Devi and Kota 1992c) and Hco M (26,48) with no restriction on ns or ng (Kuyucak et al 1991b) are shown. It is seen from figure 11 that the ng ~< 2 calculations are as good as full space calculations. Thus establishing that ngmax ,,~ 2-3 calculations are adequate for 7-unstable nuclei. Once again the sdplBM calculations consistently reproduce very well the observed E2 and E4 matrix elements; results for E2 matrix elements are given in table 3 and E4 matrix elements are given in §4.3. It should be mentioned that large number of studies with ng.... - 1 are reported in literature (Van Isacker et al 1981, 1982; Heyde et al 1983; Todd Baker et al 1985, 1989; Zuffi et al 1986; De Leo et al 1989; Sethi et al 1990, 1991; Wesseling et al 1991). m-scheme Basis: Besides using the basis (55) where L is a good quantum number it is possible to use the so called m-scheme generated by distributing particles in the fifteen single particle states shown in figure 2. Here good L states are constructed using appropriate projection method. The m-scheme method for fermions is described by Whitehead et al (1977) and the same is employed in sdg space by Kuyucak and Morrison. Though no documentation of the code is available, its existence is reported in several publications (Kuyucak and Morrison 1988a; Kuyucak et al 1991a, b). Results obtained with this code and Hco M of §2.5 for t96pt are shown in figure 11. Exact SUdo(3): An alternative to the basis defined by ns, nd and n o ((55) and the m-scheme) is to use the SU do(3) basis I{N}~(21~)kL> which is particularly useful for 448 Y Duroa Devi and V Krishna Brahmam Kota Table 3. B(E2) values for *5°Nd and ~96pt nuclei: sdolBM description. B(E2; L i --, L: )(e 2 b 2) 0IBM * Nucleus Lc-, L: Expt (lg) b) (29) ~) (39) d) 150Nd 2[ --*0[ 0-563+0.002 ") -- 0-536 0.560 4[--,2[ 0.819+0.038 .) -- 0'870 0'810 6[--*4[ 0.980+0.09 .) -- 0.945 0.883 02+ "' 21 + 0"208 __+0"009 a) -- 0"266 0"071 2 + ....~ 4 + 0"095 +_ 0'028 .) -- 0"030 0-033 2~" --* 2~+ 0'036 + 0"017"' -- 0"088 0'004 2~2 -_.,0[ 0'0024 + 0-0005 ") -- 0-002 0"012 2 ÷ ~ 2.+ 0"034 + 0"007 a) -- 0 0"073 2~ "-*0~" 0"015+0"0009 .) -- 0"010 0"012 196pt 2~" -" 0~ 0"344+0"01 b) 0'315 0"320 0"294 0.276') 0'288 + 0'014 f) 0.30 s) 22-*0[ 3 x 10 -6~' 2 x 10 -3 0"002 0"007 22+ --,21 + 0"35__+0"031f) 0'391 0"177 0"283 0"26 + 0'055 g) 41+...~ 21+ 0"403 + 0"032 f) 0"443 0'437 0'415 0"443 + 0-026 g~ 0"38 +__0"03 h) 02+ --* 2 x + 0"033 _+ 0"007 ') 0'006 0'020 0'075 0"021 _+ 0"01 h) 02 ~ 22 0"142 +_ 0"077 f) 0"448 0"299 0'177 4 2+ --*21 + 0"003 f) 0'003 0"013 0"002 0'0023 + 0'00088) 42 ~22 0"177 + 0"035 f) 0"246 0'068 0"199 0"218 + 0-043 s) 4~" --, 3 [ ~< 0"06h) 0"026 0"020 -- 42+..a, 4,+ 0"193 + 0"097 f) 0'200 0"084 0"140 0'218 + 0"054 *~ 0"18 _+ 0"09 h~ 6[ --,4[ 0"421+0"116 f) 0'500 0"359 0"450 0'494 4- 0"37 g) 0"40 + 0"1 lh) *For details of 1O (n~'" = 1), 20 (n~ "x = 2) and 30 (n~'" = 3) calculations see the respective references. a)NDS (1986); b)Sethi et al (1991); (p,p') reaction; ¢)Devi and Kota (1992b for aS°Nd and 1992c for t96pt); d)Navratil and Dobes (1991); S)Gyapong et al (1986); Coulomb excitation; f)Bolotin et al (1981); Coulomb excitation; I) Mauthofer et al (1990); Coulomb excitation; h)Cizewski et al (1978); (n, ?) reaction. well deformed nuclei. Calculations in this basis are trivial if SUing(3 ) is a good symmetry. Formulas for energies, B(E2)'s and quadrupole moments for GS band are given in (14, 15). Employing the intrinsic states corresponing to (4N, 0), (4N - 4, 2), (4N - 6, 3) and (4N-8,4),lw)=0.1 given in figure 4, Yoshinaga (1986) derived large N limit expressions for MI and E2 matrix elements in SU,do(3 ) limit; expressions for E4 sd# Interacting boson model 449 Ig6pt 2.0- + ( 1.66 ) 0+_~.~:::::=:=::~ _ _/L- (0.34) 2"-----."y~ zt ~+ ~<,, ,, - ~.+ • I I ) 2 ,_ ",.., i/ ,, -" /~/ 6 + 1.6- (0.51) 6 ~ ~ ~ (0.49) 4 + -~" / o + N. / 4 4" (o.44~ 3+ ;\..-" 3+ "'"> 1.2- (0.84) 4 + //" (o.3o) o + - -~'~ - -- -,J-,-- I,.IJ __4 + 0.8- (0.18) 4 "1" "'''~'-- (0.25) z+ __ z+ 0,4- (o.14) 2 + (0. I0) 0 + -- --¢ glBM Expt glBM (2g) (full) Figure 11. Comparison of experimental spectrum (NDS 1979; Sethi et al 1991; Bolotin et al 1981) for 196pt with sdgIBM calculations, gIBM (2g) correspond to calculations performed (Devi and Kota 1992c) employing HsyM (44) in spherical basis (55) with nrains ~ 2~ nm~xg = 2 and gIBM (full) correspond to calculations performed (Kuyucak et al 1991b) using Hcou (26, 48) in m-scheme (see § 2.6) without any truncation of the sdg space. Note that the boson number N = 6. The #-boson occupation numbers (fig)L+ obtained in gIBM (2g) calculations are given in parenthesis to the left of the corresponding levels. Calculational details are given in the respective references. matrix elements are given in § 4.1, 4.2. Using the results given in (14, 15) several well deformed nuclei are analyzed: (i) Wu (1982) analyzed the spectrum of 176Hf (it has 1 ÷ level at 1.672 MeV) by adding L2(L+ 1)2 term to (14) with fl = 0 in (14); (ii) Suguna et al (1981) analyzed ts4W (it has 3 + band starting from ~ 1.5 MeV) and 234U (it has 3 ÷ band starting from ~ 1.3 MeV) with fl = 0 in (14); (iii) Wu and Zhou (1984) analyzed 16SEr (it has 3 + band starting from ~ 1-8 MeV) by adding perturbation corrections to (14) with fl = 0; (iv) Akiyama et al (1987) analyzed XTSHf (it has K ~ = 3 + band starting from ~ 1.6 MeV but no K ~ = 1 + band/level, it also has excited K ~ = 0 +, 2 ÷ and 4 + bands occurring twice and hence interpreted in terms of (4N - 8, 4)~(w) = 0,x) and 23*U with non-zero fl in (14); (v) Ratnaraju (1986) proposed that the elementary permissible diagrams (EPD) of Bargmann and Moshinsky (1961) can be used to distinguish multiplicity labels ~ of (2/a) and they are used to write energy formulas for levels belonging to each (;L/a).The 0 + levels in ~7°yb are studied using this method; (iv) The extension of the band cut-off beyond the sdlBM value (Lma, = 2N) can be seen from (15), with (4N,0) being the lowest irrep LmaX = 4N. With this the B(E2; 450 Y Durga Devi and V Krishna Brahmam Kota L+2~ L)/B(E2;2 ÷ ~0 +) grow and come closer to rotational model values. Wu (1982) analyzed the B(E2) data for 232Th, a68Er, 17°Hf, a64Yb while Akiyama et al (1987) analyzed 238U using (15) and the data is in much closer agreement with the sdg model; (vii) Suguna et al (1981) employing (15) and its extension, fitted with a single effective charge the B(E2; 2 ÷ ~0 ÷ ) values taking r/= 2 (sd bosons) for ~2°Xe, ~I = 4 (sdg) for 198pt, lS4Gd, 186W, 158Gd, aS4pt, lS6Gd, lSSDy and 16°Gd and r/= 6 (sdgi) for 2aSRa, 228-232Th and 23°-236U and obtained good agreement. Note that for a given ~/value the ground state band corresponds to the SU(3) irrep (~/N, 0); (vii) occupation numbers (nr) take a simple form in the SU(3) limit and they are studied by Wu and Zhoh (1984) and Kota (1987). Occupation numbers (r]e~ (4N'°)c=° are N(4N + 1)/5(4N - 3), 16N(N - 1)(4N + 1)/7(4N - 1)(4N - 3) and 64N(N - 1)(2N - 3)/ 35(4N - 1)(4N - 3) for f = 0(s), 2(d) and 4(g) respectively. From these expressions it is seen that in the SUsdg(3 ) limit for the low-lying levels the g-boson occupancy is --~ 20~o. These simple calculations do not describe many important features observed in rotational nuclei, for example: (i) anharmonicity associated with the K ~ = 4 ÷ bands (E(K ~ = 4~- L = 4 +)/E(K ~ = 2~- L = 2 ÷ ) ~- 2.5 but not 2 in ~68Er); (ii) large splitting between (4N- 6, 3) K ~= 3 ÷, 1 ÷ bands; (iii) splitting between the (4N- 8,4) with = 0 and 1 irreps etc. This clearly calls for mixing calculations in SU dg(3) basis. SUsd (3) basis: Yoshinaga et al (1986, 1988) carried out careful and successful analysis of atgEr (Davidson et al (1981) using neutron capture 7-ray spectroscopy ((n,),) reaction) produced precise and almost complete information about all the bands below 2 MeV excitation in 168Er), employing SU d0(3) basis with all SU(3) irreps satisfying 2 + 2# = 4N - 3r, r = 0, 1, 2 and/~ ~< 8, 7, 6 respectively (54 SU(3) irreps, 20 distinct ones). The hamiltonian in (14) was extended to include: (i) two pieces (U and Z) that do not change the positions of W(ct)= 0 states; (ii) a piece (Hi) which does not connect the states (4N, 0)K = 0 L = 4 and (4N- 4, 2) K = 0 L = 4 and thus H 1 can produce the anharmonicity associated with the K "--4 a÷ band; (iii) a piece H 2 which does not connect (4N - 4, 2) K = 0 L = 4 and (4N - 8, 4) K = 0 W = 1 L = 4 so that mainly it lowers 02, 0~- bands; (iv) a piece P that eliminates connection between (4N- 6, 3) K = 1 L = 4 with other low-lying 4 + levels so that K = 1 band can be moved up easily. See Yoshinaga et al (1988) for the definition of U, Z, H l, H2 and P and also for details regarding the actual calculations in the SU ag(3) basis. The spectrum obtained is in excellent agreement with data as shown in figure 12. As can be seen from figure 12, sdolBM cures the problem of anharmonicity associated with the 4~- band and it also predicts correctly the position of K" = 3 ~ band. In addition, the structure ofK ~ = 0~ and 4~- bands at 1"422 MeV and 2.06 MeV are predicted to be one g-boson (~(W) = 1) and two-phonon (ct(W)=0) type respectively. The 166Er(t,p)a68Er data support the two-phonon structure of the K" = 4~- band; the TNT strength is almost zero (see Akiyama et al (1987)). The two-phonon structure is confirmed recently by Borner et al (1991) who measured the ratio R(4+)=B(E2;K~=4-~ L=4 ÷ --*2+)/ B(E2; 2+ ~ 0~s ). The experimental value is 0"52 < R(4 ÷ ) < 1.61 and sdolBM prediction is R(4 ÷) = 1-4o It should be mentioned that besides sdolBM all other models that interpret K" = 4~- band to be two-phonon type also reproduce the data but not the Solov'ev's (Nesterenko et al 1986; Solov'ev and Shirikova 1989) two-quasi particle interpretation. The calculations of Yoshinaga et al (1988) describe E2 data very well and the results for E4 data are discussed in §4.3. Experimentally there are two 1 ÷ levels and in these 1~ at 2'7MeV is a mixture of (58,3) and (56, 1) while the 12 at sdg Interacting boson model 451 G-- 4-- 3-- 6-- 4-- 2-- • 5--r. 2-- 8-- 4-- 0 6__6-- __4==. !t : --- 6-- 4-- (5 4)1 -'" (52.6) 8-- ~1~ K 4(,,,) K 2 7-- 4-- 2-~. (58,3 = 2-- 2K=3 K=O 6-- 5-- o~(56,41 4-- K=O 8- 3= (60.2) K=O o- (6o. K=2 mSEr 4-- 2-- 0-- (64.0) Energy levels K=O Figure 12. Energy levels of all positive parity bands below 2.4MeV in 16SEr. Solid lines show theoretical ones and dashed lines show the experimental band-head enegies. The label below each band-head represents an SU(3) representation (2, #) of its main component and K quantum number. States with asterisks cannot be assigned to definite SU(3) representations because the probability of their main components is less than 5070. The SU(3) label (56,4) 1 and (56,4) 2 indicates (56,4),,= o and (56,4)+=1, respectively. See figure 4 and §2.1 for the definition of W label. Note that the sdglBM calculations are carried out in the SUsdg(3) basis. Figure reproduced from Yoshinaga et al (1986) with permission from N. Yoshinaga and American Physical Society (APS). r r 0.3 k- o o o ..,,- 0.2 I VAP (I/N) 0.1 0 2 4 6 8 10 2 L Figure 13. Comparison of the sdglBM results with experiment for #-factors of the GS band members in ~66Er. Experimental data (filled circles with error-bars) is given in (Kuyucak and Morrison 1987a). The 1/N method with variation after projection (denoted by VAP (I/N) in figure,) gives #(L)= O# + 01 L(L+ 1) and shown in the figure is the curve (Kuyucak and Morrison 1987a) with go = 0-325 and 01 = - 0-0014. The SUng(3) limit gives a constant y-factor g(L)= 90; shown in the figure are the curves with g0 = 0"325 (adopted from the VAP (l/N) calculation) and go = 0.239 (obtained by best fit to data). All the #-factors are in units of PN" 452 Y Durga Devi and V Krishna Brahmam Kota 3.8MeV is predominantly (58,3). The calculated B(M1; OGs~+ 12)~1"5#N2 and E(12) ~- 3"8 MeV and they are consistent with the 1 ÷ level observed at 3.3 MeV with B(M1; 0~s--, 1~-)= 0.9 #2. Thus the state (12) usually described as isovector M1 state by p - n IBM is well described by 9IBM. However more data on B(M1) study will decide which description is correct. Finally, it should be mentioned that Arima (1985) reported similar calculations for spectra in Gd and Hf isotopes; see § 4.5 ahead. 2.7 Studies based on intrinsic states For well deformed nuclei, instead of matrix diagonalization, an appropriate approach is to construct intrinsic states (bto)N]0) for GS band by using Hartree-Bose (HB) variational method, and generate excited bands (fl, fl', 2~, 3 +.-.) by using Tamm-Dancoff approximation (TDA) (Dukelsky et at 1984; Pittel et al 1984). The HB searches for a N-boson condensate of the form IN;g~=O;s)=(Nt)-l/Z(bu)NlO)=(g!) -1/2. ~ xeob~o 10) (56) ',,/=0,2,4 where b,00 t --- s t ~ bt 2k = d~ and b 4kt = gtk" The variational parameters X:o are obtained by minimizing the expectation value of the hamiltonian (1). Note that with the asymmetry parameter ~, = 0, the CS (22) reduces to (56). The excited bands are obtained by diagonalizing the hamiltonian matrix in the TDA basis. I:K)=[(N-1)!1-1/2" b/k ? ~,tb* 01 ~N- 110) which gives the K-bands (for the fl band with K ~ = 0 ÷, orthonormaliza- tion with GS band has to be carried out), IK) = [(N - 1)!]-l/2b~(bto)N-1lO) =[(N--1)']-'/2(e~kXekb~k )(bto)N-1,O). (57) The HB + TDA equations are given in detail by Dukelsky et al (1984). It is possible (infact relatively easy as we are dealing with bosons) to carry out variation before (VBP) or after (VAP) angular momentum projection. The HB + TDA approach was applied to study E4 distributions in 156Gd and 15°Nd nuclei by Wu et al (1988) and the results are discussed ahead in § 4.4. I/N Method: Kuyucak and Morrison (1987a, b, 1988a, b,c) showed in a series of papers that energy expectation values and transition matrix elements in the states (56, 57) with angular momentum projection can be derived in analytical form to order 1/N (giving the name 1/N method) and used them to produce a number of useful results. With the hamiltonian Hq_ h in (26) and Xek defined in (56,57), the GS expectation value of H h (after angular momentum (L) projection), to order 1/N is (Kuyucak and Morrison 1987b, 1988a, b), (Hq_h)Gs L= NEo + L(L + I) } { E°-~ L(L +aN1)) + ~k=2,4xk{N(N--1)(Ak°)2+NCk+N( 1 sd9 Interacting boson model 453 Ak, = E [:'(~" + 1)]"<:O:'OlkO>ff,~,X:oXco, a = Z:(: + 1)x~0, g,~" t' ~ (2k + ]3(t:I) (k) :,X~o), 2 E. = ~ [:(~' + 1)3%X~o. (58) Similar expressions are derived for excited K = 0 (fl, ft...) and K ¢ 0 bands (Kuyucak and Morrison 1988a, b). Using a separate ansatz for x ,s that x:0 = X~o + Ye + ~22L(L + 1), variational equations for x:0 are derived and solving them will give excitation energies and X:o for various L's. In terms of the mean field coefficients X:k(L), matrix elements of transition operators are derived. For example (Kuyucak and Morrison 1988a, c; Morrison et al 1989), × ~" <:K:'OIKK>e~X},X:rX:o (59) E,~" B(M1;0~s~ 1~') =~(g3 32 (a~ N (x2°x4°) 2 (61) L(a/2) L(L+I)\ (a/2) 10]_] (62) Equation (59) gives E2 matrix elements for GS ~7 and E4 matrix elements for GS ~ K" = 4 + band. In (60-62), T M1 = gL + 9,(f ~)1 is used and 'a' is defined in (58). Equation (62) gives magnetic g-factors for GS, fl and 7 bands and (61) gives the M1 strength from 0~s to the first 1 + state (in SUsdg(3) limit the 1 + state comes from (4N - 6, 3) irrep as shown in figure 4). A number of useful results derived by using the 1/N formalism (Kuyucak and Morrison (1988a) give complete details) and the formulas given above are: (i) Using SUsag(3) limit or VBP one will not see any variation in g-factors g(L) vs L (Morrison 1986a) for GS band. However, with the above mentioned ansatz for x:0 and the expression (62) for g-factors, one sees that g(L) for GS band take the form (in sdg but not in sd) g(L) = go + gl L(L + 1). This then, as shown in figure 14, explains the observed g-factor variation in 166Er (Kuyucak and Morrison 1987a). Thus the stretching of sdg system explain the observed g-factor variation while in sdIBM one needs multi-body operators (Cohen 1987; Lipas et al 1987); (ii) The 1/N results for M1 and E2 matrix elements yield expressions for E2/M1 mixing ratios (Kuyucak and Morrison 1988c), (K+ 1) 7 1/z K=0forfl-band A(E2/M1;LK~Lg)=CkN (2L--i)~L+3)] ; K=2forT-band A(E2/M1;(L+ 1),--, L~)= C,N L( 2) (63) 454 Y Duroa Devi and 1/Krishna Brahmam Kota I I I 1 [~o] (~,4)o 7.5 I:,ii 02] (X,4)2 [404] ( k,4)4 7.0 r~,] (,,,3)1 ,.'.'>>~i / ~ [s,d'~l ;~ Es,z]~,J" 6.5 [~2] (~.2)2 \ ~ [~3z]~'l ~ [,~o]','=J' [43t] (x, lil 6.0 i,%f_<.. \ " [54 0 V= [~o] (~,o)o I I I I 0 0.1 0.2 0.3 Ilol c Figure 14. Classification of negative parity orbits in the 82-126 neutron shell (with the oscillator shell number Y = 5). In the Nilsson model the orbits are labelled by [JffnzA]fl, in the pseudo-Nilsson model they are labelled [XnzA]fl = ,~, + 1/2, where X = j~"- 1, nz = nz and A = 2f~ - A and in the SU(3) ® U(2) limit of IBFM by (2, #) KL, t) = KL 4- 1/2, where p = ,#" - nz - 1, K L = 2~ - A = A. The figure is taken from (Kota and Parikh 1991; Bohr and Mottelson 1975; Bijker and Kota 1988). Note that as N~, 2---,r/N (r/= 2 for sd and 4 for sdg) and for finite N, 2 = t/N + 4 - 2p for the orbits shown in the figure. Note that 6 is the deformation parameter (Bohr and Mottelson 1975). and the parameters Cp, C r are functions of Xgk , effective charges and 0-factors. The most extensive mixing ratio data is available for 154Gd and it is well explained by (63). The data show the predicted 1/N fall off (which indicate collective nature of these transitions); (iii) Equation (58) and its extension to other higher bands show that to order 1/N 2 all bands have same moment of inertia (MI) (Kuyucak and Morrison 1988b) even with general multipole-multipole force. It is suggested by Kuyucak and Morrison that the deviation in MI by about 20% and similar deviation in Roy = B(E2; 2; ~ 0~s) / B(E2; 2r+ --*0~s) can be taken as the signature for two quasi-particle (qp) character of 02 (fl) bands (analytical formulas for Rp~ are available). Analysis of data for MI and Rpr in rare-earth nuclei show that especially in some of the Yb isotopes the fl-bands could indeed be two qp bands; (iv) The 1/N method allows one to study triaxial shape and shape-phase transitions in sdolBM as described in §2.2. Similarly the formalism can be used to study systematics of K" = 3 +, 4 ÷ bands as discussed ahead in § 4.5 and also f14 systematics as mentioned in § 4.1. They also yield a method for deriving sdo hamiltonians as discussed in § 2.5; (v) Going beyond the simple intrinsic states given in (56, 57), expressions are derived for GS ~ 7, GS ~ fl and fl ~ ~ E2 transition strengths by incorporating band mixing sdg Interacting boson model 455 effects in the 1/N method; Kuyucak and Morrison (1989a, 1990) give details. The Mikhailov plot analysis of the expressions leads to a better understanding of glBM vis-a-vis geometric model; (vi) The 1/N method (also HB + TDA) extend easily to sdgik.., systems (i.e. system of bosons carrying angular momentum ~ = 0, 2, 4, 6, 8,... ~m~)" 3. Extensions of sdgIBM The sdglBM where no distinction is made between proton and neutron bosons is described in §§ 2.1-2.7. Two important extensions of this model are: (i) p - n sdglBM which distinguishes between proton (~) and neutron (v) bosons; (ii) glBFM where fermion degrees of freedom are included so that odd mass nuclei can be studied. The progress made in applying these two extended models is described below. 3.1 p-n sdglBM The inclusion ofn - v degrees of freedom gives rise to F-spin where Fma~ = (N, + N~)/2 and Fz = (N,- N~)/2. For low-lying states F = Fmax, while for scissors (figure lb shows the scissors mode) states F = Fma~ - 1. The p - n sdoIBM is used so far in two problems: (i) in producing oIBM-1 hamiltonians and transition operators with (N,, Nv) dependence for the parameters that define them, which facilitates study of series of isotopes or chains of nuclei; (ii) the study of scissors and other 1 ÷ states that lie outside gIBM-1 space. Studies with IBM-2 to IBM-1 Projection: The microscopic theories of sdgIBM, as already described in § 2.5, lead naturally to p - n sdglBM hamiltonians and transition operators; they do not in general preserve F-spin. Then carrying out a projection from sdglBM-2 to sdglBM-1 space, IBM-1 hamiltonians and transition operators with (N~,Nv) dependence are derived (see eq. (51) and § 4.1 ahead). They can be employed in making calculations in sdglBM1 space for describing low-lying states in nuclei which have F = Fmax. First study of this kind is due to Otsuka and Sugita (OS) (1988) who studied the properties (E~, R, S2n , Q(0~~2~-), Q(2~-) shown in figure 8) of 0~-, 2~- and 4~- levels of Sm isotopes. Starting with the simple hamiltonian (49), with Qa2 = Qa 2 ( s ) of (lO) and K ~v~4) = 0, and carrying out a simple IBM-2 to IBM-1 projection which is valid for N~ ~ N~, and employing HB + TDA approach, OS demonstrated that the sdgIBM-1 results with the projected hamiltonian and transition operators is essentially same as IBM-2 results. More important is to note that the simple calculations produce the spherical-deformed phase transition as a consequence of variation in N, (note that same set of parameters is used for all the isotopes). Similarly it is already demonstrated in §2.7, with 15°Nd as example that the IBM-2 to IBM-1 projection method is successful in describing consistently the spectra, E2 and E4 properties (for E4 properties see § 4.4); more examples are given~ by Navratil and Dobes (1991). Similarly, in §4.1, using the projection method/~4 systematics in rare-earths and actinides are well described. Another important study employing the above method is due to Engel and Beck (1991) who studied Ex(2~-), Ex(4~-), E~(22), Ex(0~-), I(0~-l[ T~2112~)[, 1(2 + II TEzII4~-)[, 1(2 + II TEzll0~-)I and [(2+ I1TezI[2~-)[ for Sn(N. = 0), Cd(N. = 1); Te(N~ = 1), Pd(N. = 2) and Xe(N. = 2) isotopes. The 456 Y Duroa Devi and V Krishna Brahmam Kota observed systematics are well described as N,, Nv effects. From these studies they conclude: "on the level of p - n sdglBM, however there is no change in the parameters, neither those of the hamiltonian nor of the electric quadrupole transition operator. Thus the sdglBM promises to be a good extension for describing the global properties of the low-lying states in even-even nuclei...." M1 States in p - n sdglBM: The scissors 1 + states in even-even nuclei, first predicted in the two-rotor model by Lo Iudice and Palumbo (1978, 1979) and first discovered in lS6Gd by Bohle et al (1984) have their simplest description in terms of p- n IBM (Van Isacker et al 1986). In p - n gIBM for deformed nuclei the 1 + states belong to the lowest mixed symmetry partition {N- 1, 1} and SU(3) irrep (r/N- 2, 1) with F-spin F = N/2 - 1, where r/= 2 for sd and 4 for sdg. The growing data on several 1 + states, the resulting M1 distributions (Richter 1988, 1991; Hartmann et al 1987), and the failure of sdIBM to explain the observed fragmentation, resulted in several studies in sdgIBM framework since the p-n gIBM accommodates much larger variety of 1 + states. In the SU~dg(3) limit of p- n gIBM the group chain and the basis states are SUsag(3)o..,u.) ® SU(2)r ~ Oag(3)KL® SU(2)). {f}={f~,f2} F=(f]- f2)/2 (64) In this limit the GS is denoted by I{N}F = N/2 (4N,0)K = 0 L= 0). The M1 operator in pn- gIBM is TMI= Z x/~{gn, p(d~dp)'+x/~gg,p(g~0o)l}, (65) p : ~',V which has four g-factors. The SU(3) and F-spin (SU(3)- F) tensorial form of the Ml-operator (65) is TM1 _ 1 ~" Fo "r'(g.o~o)l,Fo. ta A~o ~#,Fz= 0 ' A~° = (Aao. + (- 1)V°A~ov) A1 p = ~0710 (gap + 6090), A3 p = - ~-6~(gap60 - ggp); p = n, v. (66) The M1 excited states from the GS are labelled by ]{flf2}(2y,#i)KyLi;F',F'z). Note that the U~ag(15) irrep {flf2} can be symmetric {N} with F'= N/2 or mixed symmetric {N-1,1} with F'=N/2-1. The M1 excited core irreps (2y, py) are (4N - 6, 3) with F' -- N/2 (symmetric (s)) and (4N - 2, 1), (4N - 6, 3)2 with F' = N/2 - 1 (mixed symmetric (ms)); note that (4N - 6, 3) irrep appears twice for F' = N/2 - 1 and they are denoted by ~ = 0 and 1 respectively. Using the tensorial decomposition given in (66) expressions for B(M1;F = N/2(4N, O)K = 0 L---O~F'(2i, pl ) K I = 1 L I = 1) are derived (Devi and Kota 1992d) in terms of SU(3) and F-spin (SU(2)) recoupling coefficients and SU(3) isoscalar factors (ISF). Substituting the large-N limit values for the recoupling coefficients and ISF's compact analytic expressions sd9 Interacting boson model 457 are obtained, (2:, #:) = (4N - 6, 3), F' = N/2: 3~96N(hs+ )2 4x 49 3 8N.N~ 2 (2:,/t:) = (4N - 2, 1), F'=N/2-1: 4~ N (gR-) (2:,/~:)=(4N-6,3).= o, F'=N/2- 1: 4~3 96N.N~4~ (hR-)2 (2:,#:)=(4N-6,3).=~, U=N/2-1: 0 gR+ --- (N,~gR~ + N~gR,)/N, hR+ = (N,h~ + N,h,)/N, 9R- = (OR, - OR,), h R_ = (h - h,), 9Rp=7-1 {(1 +2Xo)Oap+(6--2Xo)Oop} =70-1/2(Alp--~6xoA3p), (4N + 4) (67) ho=(Oao-gao)=-~A3o; p=u,v, Xo=(4U_l~. Some comments on the results given in (67) are: (i) Wu et al (1987) adopting a somewhat different tensorial decomposition for M1 operator (unlike the one in (66)) and using SUch0(3) intrinsic states, derived the expressions in (67); (ii) in sdIBM, without g-bosons only the scissors irrep (~/N - 2, 1) with r/= 2 is excited as there will be no (33) tensor part in the M1 operator. Moreover, going from sd to sdg case there is an enhancement by a factor 2 in B(M1)'s as the corresponding strength given by (67) is of the form (3/4z~)(2(qN,)(rlN,))/(qN)(ga_)2; r/=2 for sd and q =4 for sdg. This important result is also derived by Barrett and Halse (1985) using CS formalism; (iii) the B(M1) strength to scissors (4N- 2, 1) irrep derives from both (11) and (33) tensor parts while the strengths to (4N - 6, 3) F' irreps arise due to (33) tensor part alone; (iv) there will be more fragmentation of M1 strength (compared to sdIBM) with the inclusion of g-bosons as some strength can go to (4N - 6, 3) 1 ÷ states and this fraction of the strength depends on the g-factors hp defined in (67); ho's are determined from the g-factor data of ~-band members (see (60, 62)). The hp's and go's are determined for 156Gd by Wu et al (1987), Sm isotopes by Mizusaki et al (1991) and X68Er by Kuyucak et al (1988). One can go beyond SUag(3) limit either by extending the numerical diagonalization method (both the spherical basis (55) and m-scheme defined in § 2.6) or using the intrinsic states approach discussed in § 2.7. The HB + TDA equations for pn-sdgIBM are given by Pittel et al (1984) and the extension of the 1IN method is given in detail by Kuyucak and Morrison (1989b). The 1/N method is used to describe (Kuyucak et al 1988) M1 data (~,~y and ~--,GS) in 168Er. Recently Mizusaki et al (1991) analyzed M1 strengths to 1 + states in Sm isotopes by using pn-sdgIBM (the m-scheme matrix diagonalization method is used for 14a-152Sm and HB + TDA approach for the deformed nucleus I~4Sm). The M1 excited 1 + states and the corresponding B(M1) strengths in Sm isotopes are measured by Ziegler et al (1990). The average energy of the first three 1 + states is ~3MeV and the Ei_ 1 z aB(M1;0~s ~1+) is ,-,0"4/~ 2, 0.8/~ N,2 2.2~ u 2 and 2-6/~2 for 14SSm ' ~5OSm ' 15z Sman~i 154Sm respectively. This data 458 Y Durga Devi and V Krishna Brahmam Kota is well explained by the pn-sdglBM with appropriate choice of the hamiltonian and #-factors. These calculations (Mizusaki et al 1991) also reproduce very well the #-factors for the 2~ states in 148-15¢Sm and also the g-factor variation in GS band in lS2Sm. It should be added that the hamiltonian used in these calculations suffers from the same problem mentioned earlier: it predicts excited bands (F = N/2) at energies ~ 1 MeV higher compared to the data as discussed in § 2.6. 3.2 sdIBFM The models sdoIBM-1,2 deal with even-even nuclei. In order to study odd-mass nuclei one has to couple the odd fermion to the even-even core described by sdoIBM. The resulting sdoIBFM has been studied so far by exploiting its SU(3)® U(2) limit. In the case when the odd-fermion occupies single particle natural parity orbits j = 1/2, 3/2, 5/2, 7/2,9/2 which can be split into pseudo-orbital part 2= 0, 2,4 and pseudo-spin part g -- 1/2 (with this the states of the odd-particle transform as the (40) irrep of SU(3)) and the core nucleus is described by the SU(3) limit of #IBM, two coupling schemes giving rise to SU(3)® U(2) limit of #IBFM are possible, U~@(30) ®Ur(30) ~ U~do(15) ® sun(2) ®UF(15)®UF(2) IN} {1} {f}={f,,f2} F=(fl--fz)/2 {1} I/SUsB@ (3) ® SUB(2) ® SUF(3) ® UF(2) \ ~J (2B, #,) F (40) I~I Uf~(15) ® SU"(2) ® UF(2) {f'}={f'f'2f'} F SUar(3)(;~8~,a,r) ® SUB(2) ® Ur(2) = t_FOar(3)nr(3)rL® ~= SUr(2)~ ,/2 spiy 1 ® x [ SU;(2) F, ~U~(l!,/2 ]/. (68, In chain-I the GS SUSe(3) irrep (isr,/~nr) which is often denoted by (iGs, #Gs) is determined by IBFM vs Nilsson correspondence shown in figure 14; note that for GS F = N/2. For example, for lssw, (2Gs,/~Gs) = (4N - 2, 3). However, in chain-II the determination of the GS SUBr(3) irrep is not unique. Using the C2(UBr(15)), which acts as an exchange force (Scholten and Warner 1984), the GS for lasw is given by (Kota 1986) I{N}B{1}r; {N, 1}Br(4N + 2, 1) K = 1 L= 1 d = 3/2; F = N/2). However with the IBFM vs Nilsson correspondence, the GS is I{N}B{1}r; {N, 1}nr at = 0 (4N - 2, 3) K = 1 L = 1 J = 3/2; F = N/2). Resolution of this ambiguity in determining GS description in glBFM is an important problem. So far two problems are studied in sdglBFM and they are: (i) single particle structure of rotational bands in is 5w using chain-II; (ii) M 1-distributions (in particular scissors states) for even-odd nuclei in N = 82-126 shell employing chain-I. Before addressing these problems it should be mentioned that the coupling schemes in (68) are appropriate for odd-neutron (or odd-proton) nuclei in 82-126 shell. However, for odd-proton nuclei in 50-82 shell (odd-even rare-earths) or odd-neutron nuclei in 126-184 shell (even-odd actinides) only chain-I is possible (here one should replace (40) irrep in (68) by (30) or (50) sdg Interacting boson model 459 accordingly; for upper fp-shell nuclei (40)~ (20)). In addition it is worth mentioning that chain II which is appropriate for odd-mass nuclei and the chain (64) for even-even nuclei can be embedded in a larger supersymmetric (SUSY) U(30/30) group and the corresponding SUSY group for identical bosons is U(15/30). Some aspects of U(15/30) are discussed by Yu (1988) and it has applications in other branches of physics (Dey and Dey 1990). Single particle structure in ~s5 W: Employing chain-II, the single particle structure of the low-lying rotational bands in lssw was studied by Kota (1986). The rotational bands built on [51213/2-, [510] 1/2- have nearly degenerate band heads and they form the GS bands in ~ssw. The band built on [521] 1/2- appear at ~ 1 MeV above GS. In chain-II the Nilsson orbits [51213/2- and [510] 1/2- belong to (2os,#os)= (4N+2,1) with KL=I, Kj= 1/2, 3/2 and {f'}={N,I,0}. This GS is fixed by C2(UBr(15)) as mentioned above. Similarly [52111/2- belong to (4N+4,0) and {f'} = {N+ 1,0,0}. With the above simple SUBV(3) structures for the low-lying rotational bands one can study the single particle structure of these bands. In the Nilsson model the structure of the single particle j (:) orbits is given by the amplitudes C jeff(:), C:~f) (Nilsson coupled) = E a,C~:) (69) i where Cj(:)are single particle amplitudes without mixing and a~ are the Coriolis mixing amplitudes. The IC~[~)I are calculated in chain-II using the simple formula, j(d) chain-lI- [((4N,0) L= O(40)~[[(2sv, lzsv)gg)[8 d. (70) Note that j(f) defines uniquely the pseudo-orbital angular momentum :'. The SU(3) ~ 0(3) Wigner coefficients (II > appearing in (70) are tabulated in Kota (1987). The results of chain-II are compared with the Coriolis mixing calculations of Casten et al (1972) in figure 15. Shown also are the results in the SUfff(3)® U(2) limit (Bijker and Kota 1988) and they are close to the results obtained with chain-II. Moreover, it is easy to see that the results of chain-I will be essentially identical to that of SU~r(3) ® U(2) limit. Thus the single particle structure (selection rules as can be seen from figure 15) is well described by both sd and sd# models and this is due to the goodness of pseudo-spin or equivalently suBF(3)® U(2) symmetry. In the following sub-section the SUBV(3)®U(2) limit is exploited to determine the role of g-bosons in odd-mass nuclei by studying (i) scissors states; (ii) fragmentation of M1 strength. MI states in odd-mass nuclei: Recently the question of M1 states in odd-A nuclei is being addressed and preliminary results are given for the case of a particle occupying a single j-orbit (abnormal parity orbits h11/2 or i13/2) and coupled to the core even-even nucleus, in particle-core coupling model (Raduta and Lo Iudice 1989; Raduta and De Lion 1990) and interacting boson-fermion model (Frank et al 1991). Complementing this work, scissors states arising out of the coupling of an odd particle occupying all the natural parity orbits to the (~/N - 2, 1) SU(3) irrep of the even-even core (r/= 2 for sd, 4 for sdg) are studied (Van Isacker and Frank 1989; Devi and Kota 1992e) in the SU(3)® U(2) limit (chain-I of (68)) of IBFM. The g-bosons are included 460 Y Durga Devi and V Krishna Brahmam Kota eremSuB~ (3)®U (2) I Nilsson Coupled I~m LIbel(15)® U (2) 0.8 _eft 0.6 Cj(f)[ 0.4 02 I • , 08 l-eft 0.6 ICJ(ll 0,4 0.2 I 0.8 off 0.6 CJ[/'l 0.20'4 ~/~ I I/2- 3/2- 5/2- 7/2- 9f'Z- d Figure 15. Single particle j(#) orbit amplitudes for ~8sW in the Nilsson model (Casten et al 1972), SU~r(3)®U(2) limit (Bijker and Kota 1988) of sdlBFM and Uffg(15)®U(2) limit (Kota 1986) of sdglBFM. See the selection rules for the GS K = 3/2- band which belongs to the Nilsson orbit 151213/2- ([~'1"i]3/2-) and its pseudo-spin partner [51011/2- ([41""i]1/2-) and they are separated by 24keV. There is no selection rule for the [521] 1/2-(I-420] 1/2-) and it is ~ 1 MeV from the GS. for the first time by Devi and Kota (1992e) to study the effects due to hexadecupole degree of freedom on scissors M1 states in odd-A nuclei. The question of fragmentation of M1 strength is more severe in odd-A nuclei and hence the relevance of g-bosons. All the discussion below is confined to chain-I of (68) and N = 82-126 shell nuclei. The GS in chain-I is denoted by I{N}(4N,0)®(40); (2os,#os)KL; 1/2; J:F = N/2, Fz = (N,- N,)/2) and (;tGS,#GS) follow from figure 14. The M1 excited states from the G S are labelled by I{ f' } (2', #') ® (40); (2 I, #I ) KI LI; 1/2; JI: F', F z = (N~ - N, )/2 ) and the excited core irreps (Z, #) are (4N, 0), (4N -- 4, 2), (4N - 6, 3) with F' = N/2 and (4N - 2, 1), (4N - 4,2), (4N - 6, 3) 2 with F' = N/2-1. The M1 states (2s,#s) arising out of the irrep (4N - 2, 1) are the scissor states. It is easy to work out the M1 states (2s, #I) corresponding to the core irreps { f'} (2',/z'). Ignoring the single particle part, the M1 operator in IBFM is same as the one given in (65). The single particle contributions (they depend on F-spin mixing) are negligible for the collective states under consideration (Devi and Kota 1992e). With the tensorial decomposition given in (66) the B(M1)'s take the following form, B(M1;Ji-'*Jf)= (2Ji + 1) -1 ~. "';toAF°Y ~" 2o:(2'pF°'F' ' ) (71) Fo = O, 1 20 = 1,3 sdg Interacting boson model 461 The factors X 20;(~,~,)v°;r' are expressed solely in terms of SU(3) and F-spin (SU(2)) recoupling coefficients and SU(3) ISF's (explicit formulas for X's are given by Devi and Kota (1992d)). Compact expressions for B(M1;J--*Jf)'s are derived in terms of X 20;(k'/d)F°;e' for all the allowed core irreps; 3(2j+ -1 2 0;N/2 2 (2'/~') = (4N, 0), F' =N/2: 4n 1) (gs+) (280)(XI;(4N,0))' (2'#') = (4N - 4, 2), F' = N/2: (2j+ 1)-l(hs+ )2 (240/7)(X31(4N-O'N/2 4,2)) 2 ' (2'#') = (4N - 6, 3), F' = N/2: (2j+ 1)-l(hs+ )2 (240/7)(X3;(4N-0;N/2 6,3}) 2 ' (2j 1 2 I'N/2-1 (2'/~') = (4N - 2, 1),F' = N/2-1: + 1)- (OR-) (70)(XlI(4N - 2, I)) 2 (2j - 1 2 IN/2 I 2 (2'#') = (4N - 4, 2), F' = N/2-1: + 1) (hR- ) (60/7)(X31(4N - 4,2)) (2'#') = (4N - 6, 3),= o,F' = N/2-1: (2j+ 1)-X(hR_)2 (60/7)(Xy{4 1;,N/Z-1N _ 6, 3)) 2 (2'/~') = (4N - 6, 3),= 1, F' = N/2-1: 0. (72) The gR'S and hR'S appearing in (72) are defined in (67). The location of M1 states is calculated using the energy formula (Van Isacker and Frank 1989; Devi and Kota 1992e, f) E(F;(2nI~B)®(40)(2BF#BF)KLLJM)=2N[F(8--3#BF)+I~(8--3#Bv)2] +~lL(L+l)+72J(J+l)+~t -~-F +F+I + fl[{~2 + #2 + 2B#n + 3(29 + #,)} -- {16N 2 + 12N}], (73) where F, A, 7,, ~2, ct and ~ are parameters. These are four classes of even-odd nuclei with the odd neutron in 82-126 shell that respect pseudo-spin which is a requirement for the goodness of SU(3)®U(2) limit of IBFM. The stable nuclei in these classes which can be probed by conventional (e,e'), (p,p') and (y,¢) experiments for M1 states are {lST'la9Os}, {173yb, 177~nf} an~d {'7'Yb} a_~nd their GS belong to the pseudo-Nilsson configurations [411], [413] and [420] with the corresponding suBF(3) irreps being (4N - 2, 3), (4N - 2, 3) and (4N, 2) respectively. Complete results for scissors states for these nuclei are given in figure 16. Among the nuclei listed above, 187Os is the best candidate for the study of M1 distributions as it was recently demonstrated by Warner and Van Isacker (1990) that the E2 transition strengths in this nucleus are well described by the SU(3)® U(2) limit. Therefore, as an application of the results given in (72, 73), the predictions for M1 distributions in lS7Os are shown in figure 17. The results in figure 17 show that large fraction of the M1 strength goes to the scissors states (arising out of (4N- 2, 1)ms irrep) and these states are well 462 Y Durga Devi and V Krishna Brahmam Kota ('CL) ~ NUclei ( )'./~ ) K L L d ElMer) 2 2 5/2 2 2 3/2 '! 1,,,-3,2,0 I 3/2 0 I I/2 ,il ri '-"- 2 2 512 III • Iiii 2 2 312 T • I I i I (,N-4,4) 0 2 512 - l 0 2 512 II III TIII Ilil 0 0 I/2 III III IIII Iltl GS: (,gN-2,3) I I 312 I I • I I I GS:IRN-2.3) l I 112 eTos: [510] I/2- ISgOs: [5t21 3/2- (b) [~] Nuclei ( )~, P' ) KLL J T [ t |MIV) 2 4 912 2 4 712 2 3 7/2 T[ I ('N'3'2) 2 3 5/2 2 2 512 lii oT I'' 2 2 312 1111 1i 4 4 9/2 llli, I; 4 4 7/2 Ill I Ilili Ill,! T II!l~ 2 4 9/2 T Illli I II$1! 3 3 712 GS: ( ,.?N-2.51 3 3 512 173yb: [512] 5/2- 1771"1f : [514] 7/2- (x,i,) L J (Y ['~l N.clei E(MIV) I } (~N-I.I) 312 ~3.6MM/ I 112 t b } (,,l-z.3) I ~,/2 ~2.4MeV I 1/2 !1 GS:(,~N.2) 0 0 112 rrlyb: [521] 112- Figure 16. B(M1) strengths from GS to scissors states (Devi and Kota 1992e, f). The B(M i) strength is proportional to the width of the transition lines shown in the figure. The largest transition strength is (g, - 0,) 2 x 6e where 6e = 1.53, 1-31, 3'14, 3-6 and 3"12 for 187Os, 189Os, 177Hf, 173yb and 17tyb respectively; the M1 operator used in the calculations is T ul = g,, L,, + gv L,. The location of scissors states (mean energy for a given SU(3) irrep) is indicated in the figures the spreading being ~ 15 keV for [411], [4201 nuclei and ~ 400 keV for [4~] nuclei. The dashed lines indicate a very feeble strength. The (~,/t) in the figure correspond to (2s/~¢) in the text. separated in energy from rest of the states. This feature makes it plausible that the M1 states arising out of (4N- 2, 1) irrep can be detected in experiments. Keeping this in mind complete results (figure 16) for scissors states are derived by Devi and Kota (1992e, f). 4. E4 Observables and sdglBM The hexadecupole deformation parameter f14 and its systematics across a region of the periodic table (rare-earths and actinides), the 0 GS÷ ~ 4 ?÷ transition strengths which sdg Interacting boson model 463 1,0 ] (4N 2,1)ms 0.8 1870s "3 0.6 b~ @) (4N-6.3) s 0.4 r~ Z 0.2 (4N.0)s I I I o o ,I I , I t 0 1 2 3 4 Energy (MeV) Figure 17. M1 strengthdistribution in laTOs(Devi and Kota 1992d). The parametersused in the energy formula (73) are F = 50 keV, A = 500 keV, 71 = 16 keV, 72 = 1.5 keV, ct = 200 keV and fl=-3'33keV. Similarly the o-factors (67) used are ga+=0.35 #~, ha+=0.6#N, ga_ =0"7/~N and ha_ =-0"8#s. For the final state core 0-',#') SUa(3) irreps (4N,%, (4N- 6, 3)g, (4N- 2, 1)ms and (4N- 6, 3)m~ the strengths corresponding to (2p#s) SUnr(3) irreps {(4N-2,3), (4N-4,4), (4N-4,4), (4N,2), (4N,2)}, {(4N-7,4), (4N-6,2), (4N - 6, 2) }, { (4N - 4, 4), (4N - 4, 4), (4N - 3, 2), (aN - 3, 2) } and { (4N - 7, 4), (4N - 6, 2) } respectively are shown in the figure; N = 10, N= = 3 and N, = 7. Note that strengths less than 0"01 #2 are not shown. gives information about the deformation parameter a42 and the corresponding systematics in rare-earths, several E4 matrix elements in a given nucleus (Cd-Pd region, rare-earths), E4 strength distributions (112Cd, 15°Nd, 156Gd) and excited rotational bands (K" =03, 3~-, 4~)in rare-earths give direct information about hexadecupole degree of freedom in mass A ~ 100-200 region. The analysis of this data in sdgIBM framework is carried out by several authors and the results are presented in §§4.1-4.5. These should be considered as first studies of E4 data using sdgIBM. 4.1 f14 Systematics Coulomb excitation, electron scattering and coulomb/nuclear interference has generated data on hexadecupole deformation parameter f14 all across the rare-earth region and the latest compilation of this data is due to Janecke (1981). Similarly Bemis et al (1973) employing coulomb excitation with 4He ions and Zumbro et al (1984, 1986) analyzing muonic K, L and M X-rays produced f14 data for some of the actinide nuclei. An important feature of the fig data in rare-earth and actinide region is the sign change in f14 after some value of the mass number A. As shown by Todd Baker (1985) the sdIBM fails to describe the f14 systematics in rare-earth nuclei. It is shown (Morrison 1986b; Wu et al 1988) that the inclusion of g-bosons together with the OAI mapped E4 transition operator (and using 1IN method) can ~n principle explain the change in sign of fig with A. However, as the formalism employed by Morrison (1986b) and Wu et al (1988) is strictly confined to sdgIBM-1, the 464 Y Durga Devi and V Krishna Brahmam Kota experimental data are not analyzed. For quantitative description of data, one should start with p- n sdglBM (Devi and Kota 1992g). In sdglBM-2, employing the most general quadrupole and hexadecupole transition operators, T~(uv) = E e(oa)Q~;p; 2 = 2, 4, (74) p=lC, Y determining the effective charges e~.~,.p that define Q~.o with the OAI mapping in proton, neutron spaces separately (5iJ) and then carrying out IBM-2 to IBM-1 projection (see discussion leading to (51)), the effective charges eg,t" ~) that define the IBM-1 transition operators Q~ (3,4) are obtained, eO.) ~ aO.) oO.)M / ~r C,E' = ~ ~,~';p "p ~" pll,~. (75) Writting the N-boson coherent state in sdglBM as, \ IN;~>= -¢~ M 2 = M,, = ( g; ~lQ~lg; a) = N E (ctt~r')o4e~4!. (77) 0g d,d" These CS matrix elements M 2 and M4 are related to the geometric model deformation parameters (f12,f14) for axially symmetric shapes (as is the case with SU~ag(3) and SUsag(5) limits; table 2), by the relation (Bohr and Mottelson 1975) f12 = M2 /1 4u "~M 9 f12 (78) In the geometric model analysis (Bohr and Mottelson 1975) that gives rise to (78), the surface diffuseness corrections are neglected. Note that Ro = ro A1/3 and ro is usually chosen to be 1.2 fm. The mapped E4 operator Q~ with %~(4)and e (4)~ as the two free parameters (74, 75) is used to calculate the CS matrix element M4 via (77) in the SUng(3) and SU~9(5) limits by using equilibrium shape parameters (P2,o P4,)' o o) given in table 2. The M4's thus obtained and the experimental or theoretical (using (77)) f12 values when used in (78) give f14 values. The calculated P4's are fitted to experimental data and the two sdg Interacting boson model 465 free parameters ~(4) e~) are determined by Devi and Kota (1992g) and in this analysis experimental f12 values (Raman et al 1987) are used which is equivalent to comparing the calculated and experimental E4 matrix elements (4~s It T 411 06s ÷ ). It is useful to note that the M4's are 4-8 times the single particle unit (s.p.u.; footnote to table 4 gives the definition of ~.p.u.) for rare-earths and 6-10 times for actinides (excepting in few cases). The results obtained for/~4's in rare-earth and actinide nuclei are shown in figures 18, 19. For rare-earth nuclei it is seen from the results given in figure 18a that the SU~dg(3) limit provides a good description of experimental data. In figure 18b the results of Nilsson et al (1969) are shown. Here the potential energy surfaces are constructed as a function of quadrupole (/~2) and hexadecupole (/~4) deformation parameters 0.10 "Nd ,, ' " r I 0.05 . ~40.00 Dy / 00 w v,= 12 ," .... "Os -0.05 .... %,~' "A T I I ...... "'I 'j; (o) -0,10 150 160 170 180 190 200 A "'"r--~ 0.10 0~10 . ~ , v , . ,,oooF \ A 0.05 T T -O. tO ~40 160A180 200 ~40.00 -0.05 IOVVlIQA ,xpt / f / T -0.10 150 160 170 180 190 200 A Figure 18. Hexadecupole deformation f14 vs mass number for rare-earth nuclei: (a) The SU.0(3 ) and SU do(5) predictions (Devi and Kota 1992g) are compared with data. (b) The theoretical results due to Nilsson et al (1969) are compared with data. In the inset to figure (b) the polar cap model predictions (Bertsch 1968) for the variation in M, with A is shown. The experimental data is taken from Janecke (1981). 466 Y Durga Devi and V Krishna Brahmam Kota 0.7" ~ u (o) Th 0.~ 0.0 -0.1 ...... S t.J~g(5) Cm 1 230 235 240 245 250 A 0.2 ------r (b) 0.1 0.0 0 ~;7 ° Zumbr° "t" aL (expt) t t t -0.1 230 235 240 245 250 A Figure 19. Hexadecupole deformation f14 vs mass number for actinide nuclei: (a) The SUug(3) and SU dg(5) predictions (Devi and Kota 1992g) are compared with data. (b) The theoretical results due to Brack et al (1974) and Liberl and Quentin (1982) are compared with data. The experimental data is due to Bemis et al (1973) and Zumbro et al (1984, 1986). including coulomb and pairing energies and then minimizing the potential energy, the GS deformation parameters are determined. In the inset to figure 18b, Bertsch's (1968) polar cap model prediction for M4 vs A is shown. The basic principle in this model is that the residual interaction tends to correlate particles spatially about the Z-axis and the hexadecupole moment of the resulting conical distribution of particles whose surface is at an angle 0 from the Z-axis is (with #(A)= cos 0), M4(A) oz Pc(x) dx (A) where Pc(x) is Legendre polynomial of order 4; Pc(x) = (35x 4 - 30X 2 + 3)/8. With .(" the sum of proton and neutron shell degeneracies (for rare-earths X = 76) and re(A) the number of valence nucleons (for rare-earths m(A) = A - 132), #(A) = 1 - m(A)/.Ar. It should be noted that the variation of/~4 with A is similar to that of M4(A) with A. It is seen from figures 18a, b that for rare-earth nuclei the SUsag(3 ) limit results are similar to those of Nilsson et al (1969). sdg Interacting boson model 467 For actinide nuclei in figure 19a sdglBM results and in figure 19b the HFB results of Libert and Quentin (1982) and Nilsson-Strutinsky calculations of Bracket al (1974) are shown. The SUsag(3) limit shows change in sign of f14 for Cm isotopes which is consistent with data and other theoretical calculations do not produce the sign change. Thus the SU~dg(3) limit results are somewhat better than the microscopic calculations when compared with experimental data. By comparing the SU ag(3) and SU~ag(5) results with experimental data and other theoretical calculations shown in figures 18, 19 confirm that SUug(3) limit of sdgIBM provides a good description of 134 systematics in rare-earth and actinide region (with just two free parameters). The formalism given above is extended using more realistic microscopic theory involving multi-j shell mappings to study the recently deduced B(IS4; 0cs-,+ 4 ~+ ) systematics and the results are discussed in next section. 4.2 B(IS4; O~s -, 4 + ) systematics The cq2 deformation (which includes ),-degree of freedom as can be seen from the multipole expansion for radius R(O, ?p) given in § 1 and the CS given in (22)) manifests itself in the ~GSn+ -,4 ~'+ transition strengths. In order to investigate its role, Ichihara et al (1987) measured B(IS4; 0~s-, 4~+ ) (hereafter referred to as B~(IS4)) strengths in 152'154Sm, 16°Gd, 164Dy, 166'168Er, tv6yb, 182'184W and 192OS using 65MeV polarized protons. This then provides Br(IS4) systematics across the rare-earth region. The isoscalar hexadecupole mass transition density B(IS4) is defined as Z 2 r 4 2 where m = 0 for GS-band and 2 for ),-band. With the scale factor Z/A and the assumption that mass and charge distribution coincide, the B(IS4)'s will be numerically same as (the isoscalar part of) the B(E4)'s (Govit et al 1986; Ichihara et al 1987). Assuming that the transition density p(r) in (79) has the same form (say Woods-Saxon) as the nuclear potential, and carrying out coupled channel analysis together with ),-vibrational or asymmetric rotor model, the B(IS4) values are deduced from scattering data. The fl4's deduced from B(IS4; 0cs-,4os+ + ) data are in close agreement with the data shown in figure 18. The Br(IS4) data are shown in figure 20 and they display interesting systematics. Govil et al (1987) compiled Bv(IS4) data (figure 9 of Govil et al 1987) in some rare-earth nuclei (from Sm to Pt) produced by different groups. These data which are somewhat different from Ichihara et al's (1987), are not included in figure 20 as the systematic errors in Ichihara's data will be minimal (as they study large number of nuclei all across the rare-earth region). A qualitative understanding of the Br(IS4) systematics follows from the extended polar cap model (Ichihara et al 1987) where the ~-vibration is assumed to be a superposition of particle-hole states with K~=2 ÷ near the Fermi surface (/~(A) + A#; v(A) is defined in § 4.1). Then the 0~s -, 4+ transition strength is given by #(A) + A# 2 B( IS4;O+os --'4+~r,oc P4(x)dx oC (P42(#(A)))2; (80) f,J g(A) - A# where P42(x) is associated Legendre polynomial; P42(x) = (15/2) (7x 2 - 1) (1 - x2). In the inset to figure 20b the extended polar cap model prediction for the variation 468 Y Durga Devi and V Krishna Brahmam Kota , , i , I , , , i I i , , , I , , , , I , , 4 tEr ~eExpt o oSU dg(3) ~ osdglBM ,= 3 2 to I o v 1 s~ w (o) t, o T. or I Er oo;t (RPA X 3.5) 1 o 3 Gd .~ "".. m o.o ;;o',;0 ;';0 200' I Oy ...... A I0, v 1 Sm (b) ...... ° i , , , , i , j i I i , , i i , , , i i i 50 160 70 180 190 A Figure 20. Isoscalar hexadecupole transition strength B(IS4; 0os--*4r+) vs mass number for rare-earth nuclei. (a) The SUng(3) and sdglBM (with Hsv u defined in (44)) predictions (Devi and Kota 1992h) are compared with data. The SUng(3) results with filled diamonds connected by solid lines are obtained with the effective charges e14) = 2eb 2 and e(41 = 0'1 eb 2 . Similarly, the small hollow circles connected by dashed lines are obtained with the effective charges e~ )= l'4eb 2 and e(4) = 0.3 eb 2. The results of the detailed sdglBM calculations (Devi and Kota 1992a) for 1s~zS4Sm are denoted by big hollow circles. (h) Theoretical results obtained using RPA calculations (lchihara et al 1987) are compared with data- It should be noted that all RPA results are multiplied with a factor 3"5. In the inset to figure (b) the extended polar cap model (see text) predictions for B~(IS4) vs A is shown. Experimental data with error bars is from Ichihara et al (1987). For 192Os the B~(IS4) value due to Todd Baker et al (1985) is also shown (large filled circle). of B~(IS4) with mass number A = 132-208 (appropriate for rare-earth nuclei) is shown. The observed trends are well explained by this simple model. For a more quantitative description of the data sdglBM and RPA models are used till to date and the results of these models are described below. In order to describe the above By(IS4) data, the sdglBM formalism given in §4.1 for the study of fl,~ systematics is extended to include: (i) a more realistic multi-j shell mapped (instead of single-j shell OAI) hexadecupole transition operators (T~4) in proton, neutron spaces; (ii) derivation of exact expression for E4 matrix elements connecting members of GS-band to 7-band in the SU,ag(3 ) limit (instead of using the CS formalism) and taking large-N limit. In the SUsag(3) limit the GS-band is generated by (4N,0) irrep and the y-band comes from (4N- 4, 2) irrep. Exact expression for GS-band to y-band transitions which are given by ((4N - 4, 2)K s = 2 L/= 2 II T ~ II (4N, 0)0~s) are derived (Devi and Kota 1992h) and using the large-N limit expression for the SU dg(3) recoupling sd9 Interactin9 boson model 469 coefficients (Bijker and Kota 1988) and ISF's (Wu et al 1986; Devi and Kota 1991b) analytical expressions for B~(IS2) are derived, B(ISa;O~s--*Ky = 2 Ly = 2) = 2N a°affeCe".~,(e2e'OIX2) ; re,e" ' )I a°= 7x//~; a o=~;2 aO = x/~/35; aoz = 0; a 22= x/~; a~ = x/~" (81) The structure factors a~ define the intrinsic states corresponding to (4N,0) and (4N-4,2) irreps; see the inset to figure 4. The expression (81) for 2 = 2 is same as (31c) of Yoshinaga (1986). Expression (81) for 2 = 4 is employed in analyzing the Br(IS4 ) data given in figure 20a. The effective charges eE,/'~a~ of glBM-1 E4 operator (4) are defined using IBM-2 to IBM-1 projection (75). The effective charges eg.~';p ~4~ that define T pE4 are obtained using seniority transformed multi-j shell Dyson boson mapping and are given in figure 1,2 of Navratil and Dobes (1991) for protons in Z = 50-82 and neutrons in N = 82-126 shell which are appropriate for rare-earth nuclei. These effective charges together with (81) are used in calculating Br(IS4). The results are shown in figure 20a with two sets of values for e e in (74). It is seen from figure 20a that sdgIBM description is in good agreement with data except for 168Er (i.e.) the peak in B~(1S4) data is not well reproduced. This may be partly due to the fact that the microscopic theory used to determine effective charges ~',~,';p,,,~4~ is based on seniority scheme and hence it may not be good for well-deformed nuclei. In general the departures from data can be explained by detailed spectroscopic calculations in sdg space and the corresponding results for ts2'lSgSm (Devi and Kota 1992a) are shown in figure 20b and they are in close agreement with data. On the other hand for ~92Os the results shown in figure 20a and that of detailed calculations (Devi and Kota 1992c) are essentially same and both seem to show departure from Ichihara et al's (1987) data. However, the detailed calculations are performed to explain the E4 data for the first three 4 + levels obtained by Todd Baker et al (1985) using 135 MeV polarized protons. Finally, it should be mentioned that there is also B~(IS4) data in rare-earths due to Govil et al (1986, 1987) for (X6aEr, 172yb) and Sethi et al (1990, 1991) for 194'196'19spt. The 16SEr B~(IS4) value of Govil et al (1986) is larger by a factor 2 compared to the value shown in figure 20. According to Nesterenko et aI (1988) the formula used by Govil et al has a factor 1/2 missing and hence the discrepency. They also argue that Ichihara et al's analysis has to be modified (to take into account indirect channel effects) and this gives for 16SEr B~(IS4)= 0.016e2b4. In addition to sdoIBM studies, RPA calculations (Ichihara et al 1987) in fermion space are also available for Br(IS4). In the RPA calculations the hamiltonian consists of Nilsson single particle hamiltonian, pairing interaction and quadrupole-quadrupole force. In RPA, the ~-band is generated as a two quasi-particle excitation over the Nilsson-BCS GS band. The corresponding Br(IS4) are compared with data in figure 20b. The RPA numbers are consistently smaller compared to data and to obtain agreements they are multiplied by a factor 3"5. With this, peak in the data is well reproduced but the observed trend is not reproduced in mass A > 174 region and also for 1528m. Thus a consistent description of all the B~(IS4) data shown in figure 20 calls for detailed investigations in sdoIBM and other models and this is not yet available. 470 Y Duroa Devi and V Krishna Brahmam Kota Finally, from the results given in this section together with those given in §4.1, it should be clear that the large-N limit analysis is adequate for studying gross features or systematics of E4 data. A good description of E4 data in various individual nuclei, however, calls for detailed numerical calculations, which are presented in the next two sections. 4.3 Select E4 matrix elements For significant number of nuclei E4 matrix elements involving the decay of the first few (3-6) 4 + levels are measured in the last 4-6 years. The purpose of this section is to bring together all these data and their analysis in sdglBM framework. The data are measured for vibrational Pd-Cd isotopes, rotational (~raEr, 172yb) nuclei and 7-unstable Os-Pt isotopes. Pd-Cd Isotopes: Inelastic electron scattering of lo4,106, los, 1lOpd and 1lOCd isotopes are studied (Wesseling et al 1991) in the effective momentum range 0.3-2.5 fm- 1 with energies between 70-440 MeV and the angular range of the scattered electrons covering 370-87 °. The B(E4)'s extracted from these experiments are given in table 4. They include E4 matrix elements B(E4; 0~s--.4~+ ) with i~< 2,2,6,3 and 6 for l°4Pd, lo6pd ' lOSpd ' 1lOpd and 1l°Cd respectively. The Pd-Cd isotopes are near vibrational nuclei and therefore Ua(5) limit of sdlBM is relevant. However, in the Ua(5) limit (which preserves d-boson number) no E4 transitions to GS are possible as the E4 operator in sdlBM is (did') 4. Thus to have non-zero E4 matrix elements one should have strong d-boson mixing or include explicitly g-bosons and the latter appears to be a better choice. Therefore, in order to explain the observed E4's, Wesseling et al (1991) carried out #IBM calculations (with nm"=g 1) using the hamiltonian H = Hsd + eg~g + ~[(9tst)4"(dd) 4 + h.c.] and general E2 (3) and E4 (4) operators with etal4.4- --- v,n. ~ = 2, 4 and all the free parameters are obtained by least square fit to spectra, E2 and E4 matrix elements. The results of these n o = 0 ~9 ng = 1 mixing calculations for B(E4)'s are shown in table 4. For lo4. lorpd with only two data points and three free parameters in the E4 operator there are no predictions. The effective charges (,,t4) ,,t4) ,,t41~ employed in the calculations for l°4pd, s°6pd, l°apd, 1lopd and 1SOCd' in ~04' ~22' ~24 l e fm 4 are (- 1.05, 3.30, - 10-0), (- 1.05, 3.30, - 10.0), (- 1.05, 3.30, - 9.90), (- 1.05, 3.30, - 5-40) and (1.25, 2.50, - 12"5) respectively. The calculated results for toa, ~lopd and 11°Cd are in good agreement with data and thereby demonstrating that gIBM can explain the data. The calculations show that for the 4~" states strong d-boson mixing explains the data but for other higher 4 + states inclusion of g-bosons is essential. A good understanding of the effective charges is still lacking. The authors also extract out transition charge densities in l°a'll°Pd and 11°Cd and the latter appears to be different from the former and we will return to the question of transition densities in § 5. Recently De Leo et al (1989) extracted for the first twenty eight 4 + levels in 112Cd E4 matrix elements which gives E4 strength distribution. These data together with the corresponding sdgIBM analysis is described in §4.4. Besides the above data, B(E4; 0~s --+ 4 + ) values for 4 + states at 2.22 MeV, 2"38 MeV and 2.33 MeV in 11°Cd, 114Cd and 116Cd respectively are available and the data strongly suggest (Koike et al 1969; Gillespie et al 1976; Spear et al 1977) that the above 4 + states are hexadecupole vibrational (pure lg-boson) states and in the Ud(5) x lg limit (§ 2.4) sd.q lnteractin.q boson model 471 they correspond to IN;na =0, va = O, La = 0; 9; L= 4) states. The boson number N for ~°Cd, :a4Cd and ll6Cd is 7, 9 and 8 respectively. The B(E4) values for the decay of these 4 + states to the 0~s are given by the formula B(E4]') = N(go~)2. The predicted B(E4) values for ~°Cd, ~4Cd, ~6Cd are in the ratio 0.88:i-1:1 and they should be compared with the values 0"85:1.2:1 deduced from DWBA analysis (Koike et al 1969). Thus the data is well explained by Ua(5) x 19 limit. Koike et al (1969) measured'the absolute B(E4) values; the B(E4; 0~s ~ 4~) value for ~°Cd is somewhat larger than the value given in table 4 which is due to Wesseling et al (1991). For ~6Cd B(E4)ox /B(E4) spu is determined to be 8"3 and this then determines the effective charge d0~; (e~)) z= 1.375 s.p.u. With eo ~ 2.3 MeV the energies of the 4 + levels can be reproduced in Ua(5) x 19 limit. 168Er, 172yb: For the deformed X68Er and 172yb nuclei Govil et al (1986, 1987) deduced B(IS4; 0~s ~ 4~+ ) values for four 4 + levels from the inelastic scattering data obtained using 36 MeV ct particles (for ~68Er 16 MeV deutrons are also used). These data together with data from other sources for 0gs--,4 ~ and 0,~s-~4~ in t68Er are given in table 4. For these transitions there is considerable discrepency between Govil et al's data and data from other sources as can be seen from the table. The data for 16SEr are analyzed in sdglBM framework by Yoshinaga et al (1988) employing SUrds(3) basis described in § 2.6 and by Kuyucak et al (1991c) with HcoM (§ 2.5) together with 1/N method described in § 2.7. In the Yoshinaga et al (1988) calculation, the E4 operator (4) is transformed to SUsag(3) tensors and the four effective charges are fitted to 0GS-*41,+ + 0GS--)' + 4 Y+ and vGSa + --,a _K~=4 ÷ * B(E4)'s and assuming the strength for 0as--,4 ~" is zero (no member of the B-band is seen to be excited by (~t,:() experiment (Govil et al 1986)). With these effective charges they predict B(IS4)'s + + + + + + + + for0os~4K,=0+,0os-~4K,=a+,0~s~4r=2: and0Gs~4r,=4: to be (in s.p.u.) 0"1, 50"8, 18"9 and 2"8 res~pectively. Th~ predicted sffength to 4/¢, 3+ ig in disagreement with the Govil et al's (1987) data value. They also predict ~Lll re4ll L) for 2 +, 4 +, 6 + and 8 + members of the GS band to be (in s.p.u.) -2"08, -1.31, 0-28 and 3"64 respectively. It should be added that the small B(IS4) value for the 4~,=4. is much smaller (0.6 s.p.u.) compared to other transitions and this supports the view that K~= 4 + band is two-phonon in nature (§ 2"3). In the 1/N calculations, K uyucak et al (199 l c) employ HcoM with three different parametrizations for quadrupole and hexadecupole operators (the same hexadecupole operator is used in calculating E4 matrix elements) and the results of these calculations as can be seen from table 4 are in complete variance with data. Thus the available sdgIBM descriptions of E4 properties of ~6SEr are unsatisfactory and more detailed investigations putting emphasis on E4 data (the present studies focus mainly on band structures and E2 properties) are clearly called for. Finally it should be mentioned that Jammari and Piepenbring (1990) using MPM attempted to reproduce B(IS4; 0~s ~ 4 + ), B(1S4,• 0os + ~ 4 K. + =4 +) values. With a parameter in the model these two numbers are predicted to be 0.029~e2b4 and 0.13 x 10-3 e 2 b 4 which are in good agreement with data (given in table 4). It should be mentioned that no sdoIBM calculations are available for :VzYb. However, Solov'ev and Shirikova (1989) studied the rotational bands in av2Yb using QPNM and they predict B(E4;0~s--,4~,=3+) to be 2"7 s.p.u, against experimental value of 6"9 s.p.u, given in table 4. Althougfi there are data for two K~= 2 + bands (see table 4), the authors report calculations for one K" = 2 + band and they do not give the prediction for the corresponding B(E4) value. 472 Y Dur#a Devi and V Krishna Brahmam Kota Table 4. Select E4 matrix elements in l°,,'l°6A°s'11°pd, la°Cd, 16SEr, 172yb, a92Os, 19`*'196'198pt nuclei. B(E4; L i "-* L/)(10- 2 e 2 b 2) Nucleus E(4; ) MeV L~ --* LI Expt 9IBM l°4Pd 1.324 0~ ~4~ 0.18 + 0.28 ") -- 2.082 0~ --,42 ~0-06 ") -- l°6Pd 1-229 0~- ~4~ 0.44 + 0.10 "~ -- 1"932 0~ --*42 ~0-09 ~ -- l°SPd 1.048 0~ ~ 4~ 0'70 + 0-13 ~) 0"53 ~) 1'625 0~ ~4~ ~ 0'I(P ) 0"27") 1'956 0~ ~4~ 0'41 + 0'1 ") 0-38 *) 1"990 0~ --*42 0"37 + 0" 19 ~) 0"04 .) 2'374 0~" --,4~ 1"0 + 0"2"~ 1"57~) 2'555 0~" --,4~ 0'9 _ 0'5 -) 0"02.) 11°pd 0'921 0~" ~4~ 0"91 ___0"14a) 0"65~) 1"398 01+ --*42 + ~0"10 ") 0"12 ~) 1"934 01+ ~ 43 + 0'56 +_ 0"13.) 0"55~) 11°Cd 1"542 0~ ---,4~ 0"05 +0'05 ~) 0"15 ") 1"809 01+ ~42 + 0'28 + 0'11 ") 0'70 ~) 2'220 0~ ~4~ 0'65 + 0"21.) I'0(P ) 2"251 0~ ~4~ 0'62 4- 0'16 -) 0-43 .) 2'377 01+ ....> 4 5 + -- -- 2-561 01+ ~ 4 6 ÷ 0"34 _+ 0-15 ") -- 168Er 0.264 0~- ~4; 0'4 + 3"513'9] b'*) [3.9*] f) 3.2 4- 2.0[6.6] ° [3.9*] g) 5'3_,,.7110"9]+ 7"0 d) 4.0_+6:2o [8.2]') 0.944 (y-band) 0~ ---,4~ 8'3 4- 1"6116-5] b) [16.5"] f~ 3-98 _ 0.32 [8.2] h) [0.34,0.35,0.13'] g) 1.613.29] i) 1.737 (K"=3 *) 0~--,4~ 1.8+0.513.5] j' [50.8] ° [0.59,0.56,0.16']s ) 2-055(K~=4 + ) 07~4~ 0.28+0.14[0.6] b) [0"6*] ° [0"06, 0"05, ~ 0* ]g) 172yb 0.260 0~- ---,4+ 0'03 +0"97_0.03[0"05] J) 1'263 (K"=3 +) 0~---,4~ 3"6+0-716'9] j) -- 1"658(K"=2 +) 0;~4~ 0.6+0.211.1] j) -- 1-803(K~=2 +) 0+.-,4~ ~<1.2[~<2-2]J) -- 192Os 0-58 0~- ~4~ 4"00 k) 4.71 ~) 3-84 _+. 0.4Y ~ 0.91 0 ~ --*4~ 1.39 k) 1-02 m) 1'35 _ 0'67 t) 1'07 0~ ~4~ 1.21 k) 2.25 m) 1,17') 194pt 0'811 O~ ~4~ 3"8 + 0'27 "'°): 4"22 m) 3'65 + 0'34 °) 3.31 r) 5.29 + 3.7 p'q) 1"229 O~ --,4+ 1-32 4- 0"16 ".0): 1-79 m) 1'72 + 0"16°) 2"72o 1'69 + 0"78 p'q) 1'911 O~ --*4+ 5"24 + 0"32 n'°): 4"42 m) 7'84 + 0'56 °) 1"9Y ~ (Continued) sdg Interacting boson model 473 Table 4. (Continued) B(E4; L i ---, Lf)(10- 2 e 2 b 2) Nucleus E(4f ) MeV L i --, L I Expt #IBM 196pt 0"877 0~ ~4~ 3"06 _+ 0-25 °) 3-50m) 2.40 _ 0.50 ~ 3.40 o 3.24 + 2"5p'q) 2'5ff ) 1'293 0~ ~42 2'47 + 0'28 °) 3'1 m~ 2"0 __+0-4& ) 0"80t) < 1.96 p'q) 1-99 ~) 1'537 0~ -, 4~ 0'45 +__0'08 °) 0'92 m) 1.887 0~ --,4~ 4-0 + 0.2 °) 1-99 ~) 4.4 _ 1.3 s) 4.900 2.62 r) ~gspt 0.985 0~ --*4+ 2.07 + 0'09 ") 2"16 m} 0.81 +__0.81 q~ 1.77 ~) 1'287 0~ ---*4~ 1'54 __+0"15 ") 1"54m) 0.04 ") 1'30r) 1.785 0~ --,4~ 0.88 __+0.13 ") 0.72 ~) 2,82 ~) *The numbers given in the square brackets are in single particle units (s.p.u.) B(IS4)s.p.,. = (81/196zt)(roAl/3)Se2fmS= (81/196rO(roA1/a) s x 10-Se 2 b 4 and usually r o = l'2fm is used. With this for example, for A= 110, 168, 200 and 230, B(1S4)sp=O-157 x 10-2e2b 4, 0-486 x 10-2 e 2 b 4, 0.773 x 10-2 e 2 b 4 and 1-123 x 10-2 e 2 b 4 respectively. *The number given is not a prediction but is used in fitting effective charges ~4) see text. *Three sets of parameters in the hamiltonian are used and the corresponding results for B~E4)'s are given. :The E4 data given by Sethi et al (1990) was reanalyzed by Sethi et al (1991), taking properly into account the surface diffuseness corrections. The corrected data is given in the table. In addition the E4 matrix elements measured anew by Sethi et aI (1991) are also given in the table, Kuyucak et al (199!b) analyzed the old data of Sethi et al (1990). a~Wesseling et al (1991); the authors carried out the experiment as well as sdglBM (with n m"~g = 1) analysis; b)Govil et al (1986); the 3'9 s.p.u do not follow from the B(E4) value 0.4+0.35x 10-2e2b4; see the original paper; C)Lee et al (1974); Nuclear/coulomb interference in inelastic scattering of ct particles; O)Shaw and Greenberg (1974); Coulomb excitation; C~Erb et al (1972); Coulomb excitation; fWoshinaga et al (1988); g)Kuyucak et al (1991c); hqchihara et al (1987); i)Nesterenko et al (1988); J)Govil et al (1987); E4 data for 0+~ ~4+~ ~+ is analyzed in this paper although the experiment is reported in Govil et al G~ -- =-t (1986). k)Todd Baker et al (1985); ~)Todd Baker et al (1989); the 0~s--*4 ~ matrix element is consistent with electron scattering result of Reuter et al (1984), the 0~s+ ~ 42 + matrix element is three times larger, 0~s---, 4~ matrix element is two times larger than the results from (~t, ct') experiment by Todd Baker et al (1981). ~)Devi and Kota (1991c, 1992c); ")Sethi et al (1990); °)Sethi et al (1991). P)Baktash et al (1978); Coulomb excitation: q)Gyapong et al (1986); Coulomb excitation; r)Kuyucak et al (1991b); S)Borghols et al (1985); (e,e') experiment; t)Navratil and Dobes (1991); "~Deason et al (1981); (p,p') reaction at 35 MeV. Os-Pt Isotopes: Burke et al (1978) using (~, ~') reaction for the 4 + states at 1.28, 1.63, 1.09 MeV in 18SOs, x9°Os and ~92Os respectively, Bagnell et al (1977, 1979) using 191'193Ir(t,~) 190'192Os for exciting the 4 + levels at 1.63MeV, 1-09MeV in 19°Os and 192Os respectively and Borghols et al (1985) using electron scattering for 4 + level at 1.887 MeV in 196pt have shown clearly that these levels possess hexadecupole vibrational character. These studies establish that hexadecupole degree of freedom 474 Y Durga Devi and V Krishna Brahmam Kota plays an important role in v-unstable Pt-Os isotopes and lead to several measurements of E4 matrix elements in these isotopes. Recently for (192Os, ~94pt, 196pt, 198pt) nuclei the B(E4,• 0Gs--* + 4~ + ) data for the first (3, 3, 4, 3)-4 + levels respectively became available. They are deduced from inelastic proton scattering (/~,p') experiments for 19ZOs by Todd Baker et al (1985, 1989) using 135 MeV polarized protons, for 194,198Pt by Sethi et al (1990) using 135 MeV polarized protons and for 194,196pt by Sethi et al (1991) using 647MeV polarized protons. The B(E4) data deduced from these experiments (they also determine the sign of some of the M(E4) matrix elements but they are not dealt with so far in theoretical calculations) together with the data from some older experiments are given in table 4. Todd Baker et al (1985) demonstrated that sdIBM with (d+d)4 as E4 operator fails completely in explaining the E4 matrix elements. Todd Baker et al (1985) and Sethi et al (1990, 1991) performed 19-boson calculation (n" gma~ = 1) and demonstrated that with 9-bosons one can reproduce the data. However, these calculations do not have any predictive power as the authors use three parameter E4 operator to explain three pieces of data except in 196pt. In 196pt fitting the data for 4[, 4 2 and 44 levels the data value for the 43 is well reproduced. In general to have more predictive power one has to go beyond n gma~ = 1 calculations and the results of these studies are discussed below. The isotopes ~91Os, 194pt, 196pt and 198pt are analyzed by Devi and Kota (1991c, 1992c) employing Hsw t defined in (44) by switching off H(SUd0(5)) in all the calculations and in ~gZos and 194pt calculations H!SU d0(6)) is also switched off. The truncated spherical basis with the restriction (n~ ~", nm'x] being (2, 2), (2, 2), (2, 2) 9 " and (1,2) for 192Os, 194pt, 196pt and 19Spt (with boson number N= 8,7,6 and 5) respectively is employed in these calculations. After obtaining good description of spectra and E2 properties in these isotopes (Devi and Kota 1991c, 1992c and § 2.6), E4 matrix elements are calculated using the operator that interpolates ~osc4 ~ 19v/5 ,-4 5~]]-(d,0+_,~p+3x/l~ , 4 3 Q4se:u= (stO+9*s')~+-~-(d d), 14 Y Ju 28 (9 0)uJ (82) and the Osd0(15) rank-4 generator 14 defined in (21). Thus the T E4 operator is T E4 = ~[I~ + r/Qos¢:u]4 (83) It should be noted that although Q4osg is not a generator of any of the glBM dynamical symmetries, it can be viewed as a close analogue of Q4(s) but of rank-4 (and behaves as (22) irrep of SU~do(3)), as the factors multiplying the terms in the Q4os¢are obtained by evaluating the matrix elements of r 4 y4 operator in the sdg harmonic oscillator basis and it is successfully used in earlier studies describing data relating to E4 observables (Wu et a11988; Devi and Kota 1991a). With (~, q) taking values (0-0143 eb z, - 10/3), (0.272eb 2, 0), (0.043 eb 2, 8/23) and (0.09 eb z, -2/3) for 19ZOs, 194pt, 196pt and a98pt respectively, the observed E4 data is well explained and the results are given in table 4. Note that for 19ZOs (rotor-7-unstable transitional nucleus; Casten and Cizewski 1978, 1984) Q4o5¢ is dominant while for 194pt (which is much closer to Osd(6) limit appropriate for ~-unstable nuclei; Arima and Iachello 1979) the E4 operator is essentially 14. In addition Kuyucak et al (1991b) carried out full space calculations using m-scheme matrix diagonalization and Hco w They also use a sdg Interacting boson model 475 two-parameter E4 operator and the results of their calculations are shown in table 4. Comparing with the results of Devi and Kota (1991c, 1992c), it is clear that full space calculations are not warranted for Os-Pt nuclei. For 196pt, Navratil and Dobes (1991) carried out n~ a~ = 3 calculations employing HDvs.projhamiltonian and a mapped E4 operator with proton and neutron effective charges as free parameters. Their results are in general good except for 0~s ~ 42 . On the whole it appears that sdglBM with a truncated space is capable of describing the E4 properties of 7-unstable Pt-Os isotopes. 4.4 E4 strength distributions Experimental information on E4 strength distributions (transition strengths up to ~ 2-5 MeV in deformed and -,-4MeV in vibrational nuclei) is rather scarce. In rare-earth region only for 156Gd and 15°Nd and in mass A ~ 100 region for 112Cd, data for E4 distributions are available. These data are explained so far using only sdgIBM. The data and the corresponding analysis together with the available predictions for 152'154Sm are given below. 156Gd: Goldhoorn et al (1981) measured E4 strengths in lS6Gd for excitation energies up to 2.76 MeV using inelastic scattering of 40 MeV protons. The E4 strength distribution extracted from the experimental data is shown in figure 21. A good analysis of this data is due to Wu et al (1988). They employ the hamiltonian Hq_ h in (26) with x4=0 and T2=Q2(s) of (10), ed=0.2MeV, eg=l'2MeV and x2 = - 0.002 MeV. The HB + TDA approach described in § 2.7 is adopted. The E4 distribution is constructed using E4 operator defined by the effective charges (flx ~22,+(4.) flx tt2~, fl x t~) where t t4)l,t' 's are defined in (25) and they are chosen to be those of Q4 (82). The calculated E4 distribution shown in figure 21 is obtained with the O$C choice fl = 0.7 and choosing the over all effective charge e (4~ to reproduce the observed B(E4; 0Gs~41+ + ). The gross features of the observed strength distribution and the fragmentation of the strength over many high-lying states is reasonably well described by the HB + TDA calculations but for reproducing the details of the strength distribution above 1"5 MeV more studies are required. 15°Nd: Using inelastic scattering of 30 MeV protons (p,p') E4 strength distribution in 15°Nd was measured up to 2"75 MeV by Wu et al (1988) and the data are shown in figure 22. There data are analyzed using three sdglBM calculations: (i) HB + TDA approach; (ii) matrix diagonalization with Hoys.oroj;(iii) matrix diagonalization with HoM.pror The results of these calculations are also shown in figure 22. The HB + TDA calculations are due to Wu et al (1988) and the hamiltonian and the transition operators are same as in 156Gd described above. The parameters chosen are ed =0.3MeV, eg = 0.8 MeV and x2 =-0.0026MeV in the hamiltonian and fl = 0"75 in the E4 operator with e (4) chosen to reproduce the observed B(E4; 0~s --, 4~- ). The observed strength between 1-2 MeV is overestimated by the calculations and moreover, experimentally there is sizeable strength between 2.25-2-75 MeV but the calculations predict none in this energy domain. On the whole the fragmentation of E4 strength is reasonably well explained. Navratil and Dobes (1991) employed matrix diagonalization with n max = 3 and the hamiltonian noys.proj described in § 2.5. They start with the E4 operator ge (4) Q,4 + e (4)v Qv4 476 Y Durga Devi and V Krishna Brahmam Kota x3 80 Expt 156Gd 60 4O m~ 20 N~ 0 I,, II +o x3 ,.-., 80 I HB+TDAsdg r~ 60 40 20 0 il II, 0.0 o.s ~.0 ~.s 2.0 2.s 3.0 E(MeV) Figure 21. E4 strength distributions in 156Gd. Shown in the figure are B(E4;0~s~41+) strengths summed over 0'25 MeV bins. The summed strength between 0.25 and 0.5 MeV is 288 (105 e 2 fm 8) in experiment and it is shown at 0.375 MeV as a thick bar with height equal to 288 (10~eZfma). Similarly the strength in other bins (0'5-0'75), (0'75-1'0), (1'0-1"25), (1"25-1"5) MeV etc. are shown. Note that the strength in the bin (1>25-0.5) MeV must be multiplied by a factor 3. The experimental data is from (Goldhoorn et al 1981; Wu et al 1988). The theoretical sdglBM calculations are due to Wu et al (1988) and they employ HB + TDA method. (consistent.Q2, Q4 formalism is adopted, (i.e.) the Q~ used in the hamiltonian (51) and the transition operators are same) and carry out IBM-2 to IBM-1 projection (75). The resulting IBM-1 E4 operator is used in calculating the E4 strengths and the corresponding strength distribution is shown in figure 22. The agreement between calculation and experiment is reasonable considering the fact that this is a microscopic calculation. However, the strength is underestimated roughly by a factor 2 and also the observed fragmentation between 1-3 MeV is not properly described. Employing noM.pro j described in § 2.5 and adopting matrix diagonalization approach with n smi"= 2 and n~ "x = 2, Devi and Kota (1992b) constructed E4 strength distribution for 15°Nd and the results are shown in figure 22. Here once again consistent-Q 2, Q4 formalism is adopted. This calculation although reproduces the largest strength (0os~41)+ + to be large, it underestimates the strength between 1-3 MeV. However, once again considering the microscopic nature of the calculations, one can conclude that the agreement is reasonable. Comparing the calculations described above with data it is seen that the observed E4 strength distribution in l S°Nd is reasonably well described by sdolBM although the calculations HB + TDA overestimate and HoAl.proj underestimate the strength between 1-3 MeV. sdg Interacting boson model 477 60 '150Nd x3 40 Exp~ 20 0 ili,i, 60 x3 sdg OAt 40 pro] O4 2O U3+~ oJ 0 _,i. 60 x1.5 sdg DY$ 40 proj v 20 0 II., 60 sdg x3 HB+TDA 40 20 0 - Ii i. i , 0.0 0.5 ~ .0 ~ .5 2.0 2.5 3.0 E(MV) Figure 22. E4 strength distributions in XS°Nd as measured in experiment (Wu et al 1988) and the results of sdgIBM calculations: (i) matrix diagonalization calculations with HoAi.proj denoted as sdo-OAI-proj (Devi and Kota 1992b); (ii) matrix diagonalization calculations with HDYS.proi denoted as sdg-DYS-proj (Navratil and Dobes 1991); (iii) HB+TDA calculations denoted as sdg-HB + TDA (Wu et al 1988). Calculation details are given in the respective papers. Shown in the figure are B(E4; 0~s--* 4 + ) strengths summed over 0.25 MeV bins. The summed strength between 0.25 and 0"5 MeV is 173 (10 s e 2 fm s) in experiment and it is shown at 0.375 MeV as a thick bar with height equal to 173 (105 e 2 finS). Similarly the strength in other bins (0.5-0-75), (0.75-1.0), (1.0-1.25), (1.25-1.5)MeV etc. are shown. Note that the strength in the bin (0-25-0-5)MeV must be multiplied by the factors 3, 3, 1-5 and 3 in experiment, sdo-OAI-proj, sdo-DYS-proj and sdg-HB + TDA calculations respectively. x12Cd: Using inelastic scattering of 30.7MeV and 60MeV protons De Leo et al (1989) populated twenty eight 4 + levels below 4-37 MeV in l t2Cd and deduced E4 matrix elements connecting GS to these levels. The resulting strength distribution is shown in figure 23. The authors performed nm~'= 1 calculations to describe the E4 strength distribution. The hamdtonlan chosen ts Hq_ h (26) with x4 = 0, e~ = 0.807 MeV, eg = 2"09 MeV, x2 = -0.023 MeV and T 2 operator (25) is defined by ~22t{2)= -0-56, t{2~=-0.0152,, and t~] =-0.014. The calculations predict twelve 4 + levels below 4.5 MeV against the observed twenty eight 4 + levels. The E4 strengths are calculated using the operator T E4 defined in (4) with (in 10-1 eb z units) e~o~= - 0.079, c-22,,~4) = 0.12, e~2~ = 0.6, and e~ = 0. With the fact that sdIBM cannot produce any E4 strength, 478 Y Durga Devi and V Krishna Brahmam Kota 1 1 2Cd Expt 30 20 ®~ ~o 40 sdgIBM - lg .~- 30 r~ 2O 0 1 2 3 4 E(MeV) Figure 23. E4 strength distribution in 112Cd. Shown in the figure are B(E4; 0~s--*4+) strengths summed over 0-25MeV bins. The thick bars are constructed in the same way as described in the caption to figure 21. The experimentaldata is from De Leo et al (1988) and it should be mentioned that there are twenty eight 4 + levels observed below 4.5 MeV. The theoretical sdgIBM (n~~" = 1) calculations which are denoted by sdgIBM-lg in the figure are also due to De Leo et at (1988). the sdglBM calculations are essential and the results show that lg boson calculations provide reasonable description of the observed strength (~ 60% of the strength is accounted for) and its fragmentation in the vibrational ~~ 2Cd nucleus. Predictions for 152'154Sm: Detailed spectroscopic calculations in sdg space are performed for Sm isotopes and the resulting spectra for t#s-154Sm are shown in figure 9. As already mentioned in § 2.6, these detailed calculations keep intact all the agreements shown in figure 8 for various observables and in addition they describe the fl- and y-band energies very well and also the peak seen in TNT data (figure 6). These results establish that one has good wavefunctions for Sm isotopes and therefore they can be used for making reliable predictions for E4 strength distribution for Sm isotopes, the experimental data for which is not yet available. The B(E4T)= B(E4; 0 + ~ 4~+ ) are calculated for 4 + levels below 3 MeV excitation in 152.1S4Sm and the resulting E4 strength distributions (Devi and Kota 1992a) are shown in figure 24. The Q4 defined in (82) is employed as the E4 transition operator. With the over all , osc effective charge e (4) = 0"034 eb 2, the observed B(E4; 0~s ~ 4~ ) are predicted very well; the data values are 0.14 +0.02 and 0.22 +0.02e2b 4 and the calculated values are 0"16 and 0"20e 2 b 4 for 15ZSm and ISaSm, respectively. With the same e (4) effective charge the predicted values for B(E4;0 Gs÷ ~4+~r , are 0.014, 0-014e2 b 2 for ~52Sm and sdg Interacting boson model 479 30 ,q"..~ ×6 152Sm ~ 20 iO bJ rn / I I I I I I 0 0.5 I 1.5 2 2.5 E (MeV) 30 IX8 154Sm A "{1"t-.~ 2O IO v ~1" Io l..tJ v m 0 I I L J I l 'i i i '"'I ''I " I o 0.5 , ~.~ z z s 3 3.5 E (MeV) Figure 24. E4 strength distributions in (a) 152Sm and (b) i S,Sm: The B(E4T) = B(E4; 0~s ~ 4 + ) values are shown in the figure as a function of the energy of 4~+ levels. For the 4~ level E(E4T) is to be multiplied by a factor six for IS2Sm and factor eight for 154Sm. The E4 transition operator and the corresponding effective charges are given in §4.4. 154Sm and they are in excellent agreement with the B~(IS4) data of Ichihara et al (1987) which gives 0.011 and 0.012e2b 4 respectively. With this agreement one is confident of the E4 strength distributions given in figure 24. As there is no E4 data for the 4 + levels of either GS or ? bands no predictions are yet made for 148'15°Sm. The experimental confirmation of the E4 strength distributions shown in figure 24 for ~52'154Sm will provide good test of sdglBM. The results discussed above for E4 strength distributions show that the sdglBM is a viable tool for such an analysis. Indeed, so far no other model descriptions are available for E4 strength distributions. Data for more nuclei will give a better understanding of the potentiality of the sdglBM to explain E4 data. Here it is worth mentioning that recently Fujita et al (1989) measured E4 strength distributions for (54Cr, 56Fe) and for these light nuclei the description of E4 data in sdglBM framework has to include isospin degree of freedom--this extension (Elliott 1985; Elliott et al 1987) is not yet available. 480 Y Durga Devi and V Krishna Brahmam Kota 3.0 I~)llll)lllIIIllllllllllll I]llllllfl~lIl[llllllllltl Illlllillllll;Ifi1111illll • Nd o Sm • Gd 2.5 rr_ 4- Dy • Er ~ Yb K -0 3 2.0 r -Hf -W "Os I © 1.5 P2 1.0 d v ~,.," , %. 0.5 7 Illt~llJt ILllllJll~llltlt I IIJIIl~llll[llllltllllllll IIIIIIIIIIIIIIIIIIIIl[I[ll 0.0 I i 1 I I I i I I I I I [ I I I I I I I I I I I I L illllilllillllll[lllllllll lli11111~lllIIrilIil111111 2.5 2.0 1.5 1.0 rr + 7r + rr + 0.5 K --31 K =41 K =4 2 0.0 f Ill~lllllJllIJllLJlllJlLi JlllJiJilllllllilJlillil~i 150 160 170 180 190 50 160 170 180 190 150 160 170 180 190 200 A A A Figure 25. Band-head energies of fl, 7, K" = 0~, 22+ , 31 + , 41 + and 42 bands vs mass number (A) for rare-earth nuclei. The data are from a recent compilation due to Sood et al (1991). It should be mentioned that Sood et al (1991) reported occurrence of 1 + bands in IS~Gd, l~SGd, 16°Dy, 16ZDy, l°4Dy, 166Er, 16SEt, 172yb, 174Yb, 176Hf, 182W and Is4W, at (in MeV) (1-966, 2.027), (1.845, 1-930), (2.085), (1.746), (1.746), (1.813, 1-906), (2-134, 2'365), (2-010, 2'195), (1.624), (1.672, 1.863), (2.057) and (1.613, 1.713) respectively. In addition there are also 1 + levels/bands observed ~ 3 MeV in several rare-earth nuclei which belong to scissors configuration. 4.5 Excited rotational bands Besides the GS rotational band and the one-phonon quadrupole excited fl, ? bands, in deformed rare-earth nuclei excited K"=03 , 22, 3~, 4? and 42 etc (in some cases even 1 +) rotational bands are observed. The band-head energies of these excited bands together with the fl, ? bands for rare-earth nuclei are shown in figure 25. The 03 , 2~ and 4 + bands can in principle come from two-phonon quadrupole excitation or hexadecupole vibration while the 3 ÷ band can arise only due to hexadecupole vibration. However, some of these bands can be interpreted as bands built on two quasi-particle excitations and this interpretation is outside the scope of sdglBM described in § 2 and we will return to this question at the end of the section. Thus the excited rotational bands can be understood in terms of the SU,ag(3 ) limit of sdglBM or the SUn(3 ) x 19 limit described in §2.4. In SU,a~(3 ) description as shown in figure 4 there are rotational bands (4N - 6, 3) K ~ = 3 +, I ÷ and (4N - 8, 4), = o K" = 0 +, 2 ÷ and 4 ÷ two phonon and (4N-8,4),=lK"=0 +, 2 ÷ and 4 ÷ hexadecupole vibrational and in the SUsa(3 ) x 19 limit one has %=0®(2N-8,4) K"=0 ÷, 2 ÷ sd9 Interacting boson model 481 and 4 + two-phonon and n o = 1 ® (2N - 2, 0) K" = 0 +, 1 *, 2 +, 3 + and 4 + hexadecupole vibrational bands as shown in figure 7. Though there are ambiguities in identifying the structure of excited K"= 03, 3~ and 4 + bands, the K~= 3~ band structure is relatively clean. For example, while studying the low-lying K" = 3 + bands in N = 98-104 and Z = 68-72 nuclei, Walker et al (1982) observed that in every case K ~ = 3 +, L ~ = 4 + level is excited strongly in (d,d') experiments and this coupled with the fact that (d,d') experiments excite collective states preferentially made, Walker et al (1982) conclude that K" = 3 + bands are largely hexadecupole vibrational in nature withsome two quasi-particle admixtures. In the ideal cases where the excited bands are built upon pure hexadecupole vibration (appearing around two phonon quadrupole excitation energy ~ 1.5-2 MeV) these are described by the SUd(3 ) X 1 g limit and the other rotational bands which have two- phonon admixtures can be dealt with detailed matrix diagonalization (§ 2.6) either in SUsdg(3 ) basis or spherical basis (55) or the intrinsic state approach (§ 2-7). The progress made so far in understanding excited rotational bands in sdglBM framework is discussed below. SUed(3 ) x lg description: The benchmark examples for bands built on one-phonon hexadecupole vibrations (described by SUsd(3 ) × lg limit of #IBM) come from 156Gd and 178Hf (Devi and Kota 1991a). The number of bosons is N = 12 for 156Gd, and in this nucleus a K ~ = 4 + band is observed at 1.51 MeV, in addition to the fl-band at 1"05 MeV and the ?-band at 1"15 MeV. The parameters used in constructing figure 7b reproduce the low-lying fl, 7 and K ~= 4 + bands (hereafter referred to as F-band). The structure of the F-band is IN; (2N - 2, 0); g; K = 4 L >. If needed the parameters in (35) can be fine tuned to get the precise position of K*= 4 + band _ 156Gd I EXPT i!~ 10' V///~ THEORY 2GS÷~ OGS+ ~oo N-~ 10-1 % 1o-2 +~ --I 10_3 - ITi tO-4 - El 10-5 10-6 Figure 26. Comparison between experiment (thick bars) and theory (hatched bars) for the B(E2; L+ --* Lf) values in lS6Gd for the E2 transitions from the hexadecupole vibrational K ~= 4 + band with L"= 4 +, 5 + to K"= 4 +, GS, fl and y band members. Shown also is the E2 strength from 2os+ _.~ 0os.+ The experimental data are from Van Isacker et al (1982) and the theoretical calculations are in the SUu(3) x lg limit (Devi and Kota 1991a). B(E2)'s < 10 -6 e 2 b 2 in the calculations are not shown in the figure. 482 Y Durga Devi and V Krishna Brahmam Kota vis-a-vis the/3, 7-bands and GS band. From (39, 40) it is easily seen that E2 decay of the F-band members to the GS band is forbidden (as AK > 2) and so also the decay to the/3-band. Similarly from (38, 40) one can see that the intraband transitions (F ~ F) are strong and go as N 2 while the transitions to 7-band are independent of N. Thus F--* F >> F ~ >> F--*/3, GS; the later two being zero. This trend is clearly seen in the data given by Van Isacker et al (1982) and these authors have to perform elaborate numerical (lg-boson) mixing calculations to demonstrate that the sdg model predicts the same, while the analytical results given in § 2.4 bring out the above feature instantaneously. Experimental B(E2) values (in units of 10-Ze2b 2) for 4~ ~ L + ~ 1, Y 4~- ~GS, /3~0, 5r ~ L + ~ 2, 5~ ~GS~0 and 5~" ~4~ ~ 150. The data are well explained as shown in figure 26. Using T E2 defined in (36) and fixing the value of % by using the experimental (in units of 10-2e/b 2) B(E2) value 91 for 2~s--*0~s, the 5(- ~4~- decay strength turns out to be 126 against the observed value 154. With a reasonable value for ~24,(2) in (36) the strengths going to y-band members are obtained to be close to experimental values. The nucleus ~TSHfis subjected to an extensive study (via (n, y) reactions) by Hague et al (1986). The number of bosons N = 15 for 178Hf. In this nucleus, in addition to the/3 band at 1"20 MeV and y-band at 1"17 MeV, also observed are the K"= 0 +, 4 +, 2 +, 3 + bands (hereafter referred to as Fo,F4,F2 and F3) at 1.43, 1.51, 1.56 and 1-76 MeV respectively. In the sdIBM description (Hague et al 1986), the Fo, Fz and F4 bands come from (2N-8,4) irrep, while the F 3 band necessarily belongs to SU~a(3 ) x lg. The possibility of describing the Fo, ['2 and ['3 bands as SUsd(3 ) x lg bands is investigated by Devi and Kota (1991a). Assuming the Fo band to be IN; (2N - 2, 0); g; 0L >, the B(E2) values are calculated (only a few experimental numbers are available) and are compared with data in figure 27. The B(E2)'s for 2ro--~0GS,+ + 2~s, Or+ compare quite favourably with theory, the 2ro+ --' 0~-o strength is stronger by 178H f ~ EXPT 105 lY////l THEORY ,o3 .2.0 io.2 Figure 27. Comparison between experiment (thick bars) and theory (hatched bars) for the B(E2) ratios B(E2; Li~ LI)/B(E2; L'i~ Lf) in 17SHf for the hexadecupole vibrational K It --03 + (F0) and K"= 3~ (Fa) band members. The experimental data are due to Hague et al (1986) and the theoretical calculations are in the SU e(3) x lg limit (Devi and Kota 1991a). B(E2) ratios < 10 -2 in the calculations are not shown in the figure. sd9 lnteractin9 boson model 483 a factor 100 as expected in theory. However the B(E2)'s for 4ro--+2as,+ + 3 + are not consistent with theory (here one is comparing very small numbers). In the absence of 4 ro+ -O2~o strength (and in general Lro+ ~(L ' )ro) + it is difficult to infer the structure of the Fo-band. This together with the absence of data for intraband transitions for l-'4-band as well makes it plausible that the F o, 1-2 and ['4 bands are mixtures of the sdlBM irrep (2N - 8, 4) and the 19-boson bands iN; (2N - 2, 0); 9; K = 0, 2, 4 L >. It is seen that 4 r3+3r3÷ + is larger by a factor of thousand compared to the 4 F3÷ --+5y + ' 3 + and the strength to 2~s is almost zero. Using (38-40) the data is easily explained + ~ + (figure 27). In the theory 4r3 2GS is zero, 4; --,5 + ' 3 7+ is independent of N while 4+r~3~ ~N2' Thus it is qaite plausible that the F3 band is IN; (2N-2,0); 9; K = 3L > and this is consistent with the established E4 character of this band (Nesterenko et al 1986). It is worth mentioning that in the past the F3 was described in terms of the (4N - 6, 3) irrep of the SUing(3 ) group (Akiyama et al 1987). In the N - ~:, limit (figure 4), the essential structure of the (4N - 6, 3)K" = 3 + band is same as that of the above SUsd(3) x 19 F3-band. SUing(3 ) and mean-field descriptions: Excited rotational bands in 16SEr are studied using truncated SU~a0(3) basis by Yoshianaga et al (1986, 1988) as described in §2.6 and the spectrum is shown in figure 12. They obtain very good description for the E2 properties of the various bands. From the structure of these bands as given by the SU~ao(3) irrep with largest amplitude (see figure 12) it is clear that K " = 03 + band is hexadecupole vibrational in nature ((56, 4), = 1), K" = 3 ~- band is also hexadecupole vibrational in nature ((58, 3)) and K" = 4~ band is two-phonon in nature ((56, 4),=o). Thus detailed SU d0(3) studies yield clear information about the structure of rotational bands. In addition to 168Er, the nuclei ~S6Gd and 176Hf are also studied employing SU d0(3) basis and only the spectra for the latter two nuclei are reported (Arima 1985). The detailed SUing(3) calculation confirms the hexadecupole vibrational nature of the K n = 41 band in ~56Gd. Thus validating the SUn(3 ) x 19 description for this band which is discussed in detail above. In ~76Hf (N = 16) the K"=03 band at 1.293 MeV is shown to be predominantly (56,4),= o and the K" = 3~- band at 1"578 MeV belongs to (58,3) irrep. Thus the K~= 03 band in this nucleus is two-phonon in nature while the 31 band is hexadecupole vibrational in nature. Using the mean field approach and the hamiltonian H n(26), Kuyucak et al (1991c) carried out model studies for the systematics of the band head energies and their E2 and E4 properties. Here the consistent quadrupole and hexadecupole formalism is used where the E2 and E4 transition operators are same as the quadrupole and hexadecupole operators in the hamiltonian (i.e. T 2 and T 4 in (26)). The model studies are carried out for different choices of ~(,t",~k) . k = 2, 4 that define T k in (25). Here the T 2 operator is chosen either with (flx ~22'.(2) fl X ,(2)'24., fl X ~(2)'1"44, referred to as (~,/3,/~) × SU(3) or (~ x ~22,'(~) t~2~, ~ x °44,'(2qreferred to as (~, 1,/3) x SU(3), where -22,t(2) t(2)~24 and -44t~2~ are the ones corresponding to (1/4) ~ Q2(s) defined in (10). For each of these choices of T 2, the T 4 operator is chosen to be (/3 x t~2~,/3 x t~, /3 x t~) referred to as (/3,/3,/3) x Q4osc or (flx tt2~, t24'.,4) /3 x t~) referred to as (/3, 1,/3) x Q4os¢ where t~, t~2~ and t~ correspond to Q4o defined in (82). In addition, given a T 2 operator a consistent T 4 operator can be defined using (48) and this is referred to as Tco4 u. Kuyucak et al (1991c) studied the energies of/3, ~, 03+ , 2 2 + , 31, 4 + and 1 + band-heads with various combinations of T 2 and T 4 operators defined above. With the choice ea =0MeV, K2 m --0.02 Mev', N = 16 and varying /£4//£2 from -0-4 to 484 Y Durga Devi and V Krishna Brahmam Kota 0.4 performed model calculations at the grid points defined by eg =0 MeV, 0.75 MeV, 1.5 MeV and //= 0"5, 1. They conclude that the choice T2= (fl, 1,fl)x SU(3) and T 4 = Tco 4 M is appropriate for reproducing the observed relative band-head positions. For making realistic calculations, the approximate values for the parameters can be read off from the plots given by Kuyucak et al (1991c). Systematic analysis of the existing data shown in figure 25 is not attempted so far but some sample examples are given in Kuyucak et al (1991c) for 15°Nd, XS6Gd and X6SEr. For example, for 15°Nd with eg=0.5MeV.. T2=(fl, l,fl) xSU(3) with fl=2 and T 4= TcoM,4 x 2 = -- 6"5 keV and x4/'x2 = 0.4, the band-head energies of fl, 7, K~ = 3~- and K ~ = 4~- are determined to be (in MeV) 0"678, 0"823, 1.105 and 1.466 and the experimental values (Kuyucak et al 1991c; references therein) are 0.676, 0.93, 1.06 and 1.28 respectively. In this description it follows from (57) that the K~= 3 ÷ and 4 + bands are hexadecupole vibrational in nature as two-phonon structures are not included. It should be noted that these studies can also be carried out using the matrix diago- nalization approach in spherical basis (55) and the structure of the bands can be inferred from the analysis of s, d and # orbit occupation numbers, two nucleon transfer studies and by extracting the SUsag(3) content of the wavefunctions. The studies described above show that sdglBM provides a good description of various excited rotational bands in XS°Nd, 156Gd, X68Erand x76,17SHf " Several bands in these nuclei are hexadecupole vibrational in nature and some of them are of two-phonon "type. However, they may also contain significant two quasi particle admixtures. (Nesterenko et al 1986; Solov'ev and Shirikova 1981, 1989) conclude from their theoretical analysis where they employ QPNM that K" L" = 3~ 4 ÷, 4~ 4 + levels in Gd and Dy are essentially of two quasi particle type, in Er, Yb and Hf they are mixture of two quasi particle and hexadecupole vibrational while in W and Os they are predominantly of hexadecupole type. For a complete understanding of the structure of these excited rotational bands one has to include two quasi particle excitations in sdolBM, (i.e.) to extend sdolBM to sdolBFFM where two fermions are coupled to sdo core; this extension enables one to confront Nesterenko et ars (1986) results. However, this important extension is not yet investigated in any detail. 5. Conclusions and future outlook The extended sdglBM, whose algebraic structure gives rise to a large number of analytical results, together with the numerical diagonalization and self-consistent mean field methods developed in detail, has been demonstrated to be a viable tool for analyzing and predicting E4 properties of nuclei. The model encompasses many new features that arise due to hexadecupole degree of freedom. The p-n sdglBFFM which includes hexadecupole vibrations, two phonon excitations and quasi particle structures with proton-neutron degrees of freedom appears to be the ultimate model for complete understanding of band structures and associated E4 properties of heavy even-even nuclei. Much of the future of the sdg model depends on the development of this extension, which will enable us to study high spin states, superdeformed bands, odd-odd nuclei etc. A fermionic model that accommodates sd9IBM symmetries provides deeper insight into the sdg model. One plausible approach to this problem is to extend the SD Ginocchio model (Ginocchio 1980) that accommodates sdlBM symmetries to the SDG case. The corresponding sdg Interacting boson model 485 so called k-active and/-active schemes with k = 2 (Pan et al 199 I) and i = 5/2 (Okai et al 1990) that generates some of the sdg dynamical symmetries received attention recently. Another important question that arises out of the success of sdglBM is about the relevance of i (~ = 6)-bosons. Some preliminary studies on the classification of the dynamical symmetries of sdgi model which accommodates exceptional Lie- groups and their applications (Morrison et al 1991) and also results in the SU,dgi(3) limit for deformed nuclei (Suguna et al 1981; also Chen 1991) are available. In addition to all the E4 data described in § 4, the transition charge densities or form factors for exciting 4 + levels are measured in Os-Pt isotopes (Burke et al 1978; Bagnell et al 1979; Boeglin et al 1988; Todd Baker et al 1989; Sethi et al 1990, 1991), Pd-Cd isotopes (De Jager 1986; Wesseling et al 1991) and l°°Ru (Sirota et al 1989) etc. The transition densities determine in a transparent way the hexadecupole character of the corresponding 4 ÷ levels. By extending the E4 operator in (4) where the effective charges are replaced by functions that depend on radial coordinate, some attempts are made (Wesseling et al 1991) to analyze the above data and more detailed studies are clearly called for. Besides all the observables that are discussed so far that give direct information about hexadecupole degree of freedom, there are several observables that give indirect information and they are: (i) fragmentation of single particle strength as seen in W isotopes (Casten et al 1974); (ii) the spacing between Nilsson orbits [514] 9/2- and [404] 7/2 ÷ in Lu isotopes and similarly in some other deformed odd-mass nuclei (Jain et al 1990); (iii) signature inversion in 176Ho (Xing and Xie 1990); (iv) sub-barrier fusion cross-section (Fernandez et al 1991; Balantekin et al 1991) etc. Understanding of these observables in sdglBM/sdglBFM may give new insights. A quite different usefulness of sdglBM is in understanding collective effects on statistical (average) properties of nuclei. Recently some progress has been made in applying sdlBM with temperature dependent boson number (Maino et al 1990) to address questions in statistical nuclear physics; also available are trace propagation methods that give moments of densities in boson spaces (Kota and Potbhare 1980, Kota 1981). In addition, sdg hamiltonians interpolating the syrhmetries, shown in figure 4, will enable one to learn about some aspects of order to chaos to order transitions, and the CS given in § 2"2 will give their classical analogues. Some of these questions in "quantum chaos" have already been addressed in sdlBM framework (Alhassid et al 1990; Alhassid and Whelan 1991; Mizusaki et al 1991). In conclusion, the progress made in developing and understanding sdglBM as described in § § 2, 3 and the successful description of E4 data given in § 4, should be considered as the end of the begining in understanding hexadecupole degree of freedom in nuclei. In the last three decades, a great deal was learnt about quadrupole degree of freedom and the focus brought in this article on E4 properties should stimulate many future experiments looking for hexadecupole effects. We are poised to learn a great deal about hexadecupole collective mode in nuclei in the next decade. Acknowledgements The authors express their deep sense of gratitude to F Iachello, S P Pandya and J C Parikh for their encouragement in all the years in which the work on sdglBM was carried out. Special thanks are due to F Iachello for many useful discussions, 486 Y Durga Devi and V Krishna Brahmam Kota for his interest in the subject and valuable suggestions and comments received from time to time. Some of the work reported here was carried out during the visits of one of the authors (VKBK) to University of Rochester and he is thankful to J B French for hospitality at Rochester. Collaborations with D G Burke and J A Sheikh are acknowledged. The authors are benefitted from discussions and/or correspondence with D Cline, J P Elliott, K Heyde, A K Jain, R S Nikam, T Otsuka, A Sethi, B R Sitaram, P Van Isacker and N Yoshinaga. Thanks are due to D Majumdar for his help in preparing the manuscript. References Akiyama Y 1985 Nucl. Phys. A433 369 Akiyama Y, Heyde K, Arima A and Yoshinaga N 1986 Phys. Lett. B173 1 Akiyama Y, von Brentano P and Gelberg A 1987 Z. Phys. A326 517 Alhassid Y, Novoselsky A and Whelan N 1990 Phys. Rev. Lett. 65 2971 Alhassid Y and Whelan N 1991 Phys. Rev. Lett. 67 816 Allaart K, Boeker E, Bonsignori G, Savoia M and Gambhir Y K 1988 Phys. Rep. 169 209 Amos K, Chen X J, Dieperink A E L and Morrison 1 1989 Aust. J. Phys. 41 633 Arima A 1985 in Nuclear structure (eds) R Broglia, G Hagmann and B Herskind (Amsterdam: North Holland) pp. 147 Arima A and Iachello F 1975 Phys. Rev. Lett. 35 1069 Arima A and Iachello F 1976 Ann. Phys. (NY) 99 253 Arima A and Iachello F 1977 Phys. Rev. C16 2085 Arima A and Iachello F 1978a Phys. Rev. Lett. 40 385 Arima A and Iachello F 1978b Ann. Phys. (NY) 111 201 Arima A and Iachello F 1979 Ann. Phys. (NY) 123 468 Arima A and Iachello F 1981 Ann. Rev. Nucl. Part. Sci. 31 75 Arima A and Iachello F 1984 Adv. Nucl. Phys. 13 139 Bagnell R D, Tanaka Y, Sheline R K, Burke D G and Sherman J D 1977 Phys. Lett. !166 129 Bagnell R D, Tanaka Y, Sheline R K, Burke D G and Sherman J D 1979 Phys. Rev. C20 42 Baktash C, Saladin J X, OBrien J J and Alessi J G 1978 Phys. Rev. C18 131 Balantekin A B, Bars I and Iachello F 1981 Nucl. Phys. A379 284 Balantekin A B, Bennett J R and Takigawa N 1991 Phys. Rev. C44 145 Bargmann V and Moshinsky M 1961 Nucl. Phys. 23 177 Barrett B R and Halse P 1985 Phys. Lett. B155 133 Bemis Jr C E, McGowan F K, Ford Jr J L C, Milner W T, Stelson P H and Robinson R L 1973 Phys. Rev. C8 1466 Bertsch G F 1968 Phys. Lett. B26 130 Bes D R, Broglia R A, Maglione E and Vitturi A 1982 Phys. Rev. Lett. 48 1001 Betts R R and Mortensen M H 1979 Phys. Rev. Lett. 43 616 Bhatt K H, Parikh J C and McGrory J B 1974 Nucl. Phys. A224 301 Bijker R and Kota V K B 1988 Ann. Phys. (NY) 187 148 Bjerregaard J H, Hansen O, Nathan O and Hinds S 1966 Nucl. Phys. 86 145 Boeglin W, Egelhof P, Sick I, Cavedon J M, Frois B, Goutte D, Meot V, Leconte P, Phan X H, Platchkov S K, Williamson S and Girod M 1988 NucL Phys. A477 399 Bohle D, Richter A, Steffen W, Dieperink A E L, Lo Iudice, Palumbo F and Scholten O 1984 Phys. Lett. B137 27 Bohr A 1951 Phys. Rev. 81 134 Bohr A 1952 Mat. Fys. Medd. Dan. Vid. Selsk. 26 14 Bohr A and Mottelson B R 1953 Mat. Fys. Medd. Dan. Vid. Selsk. 27 16 Bohr A and Mottelson B R 1975 Nuclear structure (Massachusetts, Reading: Benjamin) Vol II Bohr A and Mottelson B R 1980 Phys. Scr. 22 468 Bohr A and Mottelson B R 1982 Phys. Scr. 25 28 Bolotin H H, Stuchberry A E, Morrison I, Kennedy D L, Ryan C G and Sie S H 1981 Nucl. Phys. A370 146 sdg Interacting boson model 487 Borghols W T A, Blasi N, Bijker R, Harakeh M N, De Jager C W, Van Der Laan J B, De Vries H and Van Der Werf S Y 1985 Phys. Lett. B152 330 Borner H G, Jolie J, Robinson S J, Kruche B, Piepenbring R, Casten R F, Aprahamian A and Draayer J P 1991 Phys. Rev. Lett. 66 691 Brack M, Ledergerber T, Pauli H C and Jensen A S 1974 Nucl. Phys. A234 185 Broglia R, Hansen O and Riedl C 1974 Adv. Nucl. Phys. 6 287 Brussaard P J and Glaudemans P W M 1977 Shell model applications in nuclear spectroscopy (Amsterdam: North Holland) Burke D G 1990 (private communication) Burke D G, Davidson W F, Cizewski J A, Brown R E, Flynn E R and Sunier J W 1985 Can. J. Phys. 631309 Burke D G, Shahabuddin M A M and Boyd R N 1978 Phys. Lett. 1378 48 Castanos O, Draayer J P and Leschber Y 1987 Ann. Phys. (NY) 156 41 Casten R F, Kleinheinz P, Daly P J and Elbek B 1972 K. Dan. Vidensk. Selsk. Mat-Fys. Medd. 38 No. 13, 1 Casten R F, Breitig D, Wasson O A, Rimawi K and Chrien R E 1974 Nucl. Phys. A228 493 Casten R F and Cizewski J A 1978 Nucl. Phys. A309 477 Casten R F and Cizewski J A 1984 Nucl. Phys. A425 653 Casten R F and Warner D D 1988 Rev. Mod. Phys. 60 389 Chacon E, Hess P O and Moshinsky M 1987 J. Math. Phys. 28 2223 Chacon E, Hess P O and Moshinsky M 1989 J. Math. Phys. 30 970 Chakraborty M, Kota V K B and Parikh J C 1981 Phys. Lett. BI00 201 Chen H T 1991 Chin. J. Phys. 29 411 Chen H T, Kiang L L, Yang C C, Chen L M, Che~ T L and Jiang C W 1986 J. Phys. G12 L217 Chen H T, Richardson R W, Kiang L L, Tzeng Y, Teng P K, Kiang G C, Wong C W, Tsai S F, Lin E K and Arima A 1988 J. Phys. G14 L205 Cizewski J A, Casten R F, Smith G J, Macphail M R, Stelts M L, Kane W R, Borner H C and Davidson W F 1978 Phys. Rev. Lett. 40 167 Cizewski J A, Flynn E R, Brown R E and Sunier J W 1979 Phys. Lett. B88 207 Cohen T D 1987 Phys. Rev. Lett. 58 605 Cooper T, Bertozzi W, Heisenberg J, Kowalski S, Turchinetz W, Williamson C, Cardman L, Fivozinksy S, Lightbody Jr J and Penner S 1976 Phys. Rev. C13 1083 Daley H J and Iachello F 1986 Ann. Phys. (NY) 167 73 Davidson W F, Warner D D, Casten R F, Schreckenbach K, Borner H G, Simic J, Stojanovic M, Bogdanvic M, Koicki S, Gelletly W, Orr G B and Stells M L 1981 J. Phys. G7 455 Deason P T, King C H, Ronningen R M, Khoo T L, Bernthal F M and Nolen Jr J A 1981 Phys. Rev. C23 1414 Debenham P and Hintz N M 1972 Nucl. Phys. A195 385 De Jager C W 1986 in Nuclear structure, reactions and symmetries (eds) R A Meyer and V Paar (Singapore: World Scientific) vol. 1 pp. 404 De Kock E A and Geyer H B 1988 Phys. Rev. C38 2887 De Leo R, Blasi N, Micheletti S, Pignanelli M, Borghols W T A, Schippers J M, Vander Werf S Y, Maino G and Harakeh M N 1989 Nucl. Phys. A504 109 Devi Y D 1991 Some studies in sdg interacting boson model of atomic nuclei Ph.D. Thesis (Gujarat University, India) Devi Y D, Kota V K B and Sheikh J A 1989 Phys. Rev. C39 2057 Devi Y D and Kota V K B 1990a Fortran pro#rams for spectroscopic calculations in (sdg)-boson space: The package SDGIBMI (Physical Research Laboratory Report PRL-TN-90-68) Devi Y D and Kota V K B 1990b Z. Phys. A337 15 Devi Y D and Kota V K B 1991a J. Phys. G17 465 Devi Y D and Kota V K B 1991b J. Phys. G17 L185 Devi Y D and Kota V K B 1991c Proc. Syrup. Nucl. Phys. (Bombay, India) B34 47 Devi Y D and Kota V K B 1992a Phys. Rev. C45 2238 Devi Y D and Kota V K B 1992b A simple microscopic sdglBM description of spectrum, E2 transition strengths and E4 strength distributions in 1~0Nd (Physical Research Laboratory report PRL-TH-92/13) Devi Y D and Kota V K B 1992c sdglBM description of E4 Matrix elements in ~-unstable Pt-Os isotopes (Physical Research Laboratory report PRL-TH-92/14) Devi Y D and Kota V K B 1992d Phys. Lett. B287 9 Devi Y D and Kota V K B 1992e Nucl. Phys. A541 173 488 Y Durga Devi and V Krishna Brahmam Kota Devi Y D and Kota V K B t992f in Group theory and special symmetries in nuclear physics (eds) J P Draayer and J Janecke (Singapore: World Scientific) pp. 122 Devi Y D and Kota V K B 1992g Phys. Rev. C46 1370 Devi Y D and Kota V K B 1992h B(IS4; 0+~o~4 + ) Systematics in rare-earth nuclei: SUng(3) description (Physical Research Laboratory report PRL-TH-92/lS) Dey J and Dey M 1990 Phys. Lett. 11234 349 Dieperink A E L and Wenes G 1985 Ann. Rev. Nucl. Part. Sci. 35 77 Dobaczewski J and Sskalski J 1988 Phys. Rev. C38 580 Dobaczewski J and Sskalski J 1989 Phys. Rev. C40 1025 Draayer J P 1991 Capture gamma ray spectroscopy (ed) R W Hoff (New York: AIP Conference proceedings 238) pp. 30 Draayer J P and Weeks K J 1984 Ann. Phys. (NY) 156 41 Druce C H, Pittel S, Barrett B R and Duval P D 1987 Ann. Phys. (NY) 176 114 Dukelsky J, Dussel G G, Perrazo R P J. Reich S L and Sofia H M 1984 Nucl. Phys. A425 93 Dukelsky J, Niello J F, Sofia H M and Perrazo R P J 1983 Phys. Rev. C28 2183 Edmonds A R 1974 Angular momentum in quantum mechanics (Princeton: Princeton University Press) Eisenberg J M and Greiner W 1987 Nuclear models: collective and single-particle phenomena (Third edition) (Amsterdam: North-Holland) Elliott J P 1958a Proc. R. Soc. (London) A245 128 Elliott J P 1958b Peoc. Soc. (London) A245 562 Elliott J P 1985 Rep. Prog. Phys. 48 171 EUiott J P, Evans J A and Williams A P 1987 Nucl. Phys. A469 51 Endt. P M 1979a At. Data Nucl. Data Tables 23 3 Endt P M 1979b At. Data Nucl. Data Tables 23 552 Engel O and Beck F 1991 Phys. Lett. B271 1 Erb K A, Holden J E, Lee I Y, Saladin J X and Saylor T K 1972 Phys. Rev. Lett. 29 1010 Fernandez Niello J O, di Tada M, Macchiavelly A O, Pacheco A J, Abriola D, Elgu M, Etchegoyen A, Etchegoyen M S, Gil S and Tessoni J E 1991 Phys. Rev. C43 2303 Frank A, Arias J M and Van Isacker P 1991 Nucl. Phys. A531 125 Flank W and Lipas P O 1990 J. Phys. G16 1653 Fujita Y, Fujiwara M, Morinobu S, Katayama I, Yamazaki T, Itahashi T, Ikegami H and Hayakawa S I 1989 Phys. Rev. C40 1595 Garrett P E, Burke D G, Devi Y D and Kota V K B 1990 Book of abstracts (Symposium in Honour of A. Arima, Santa Fe, New Mexico) pp. 57 Geyer H B 1986 Phys. Rev. C34 2376 Gillespie A W, Singhal R P, Brain S W, Johnston A and Lees E W 1976 J. Phys. G2 185 Gilmore R and Draayer J P 1985 J. Math. Phys. 26 3053 Ginocchio J N 1980 Ann. Phys. (NY) 126 234 Goldhoorn P B, Harakeh M N, Iwasaki Y, Put L W, Zwarts F and Van Isacker P 1981 Phys. Lett. B103 291 Goodman A L 1979 Adv. Nucl. Phys. ll 263 Govil I M, Fulbright H W and Cline D 1987 Phys. Rev. C36 1442 Govil I M, Fulbright H W, Cline D, Wesolowski E, Kotlinski B, Backlin A and Gridnev K 1986 Phys. Rev. C33 793 Griffin R E, Jackson A D and Volkov A B 1971 Phys. Lett. B36 281 Gyapong G J, Spear R H, Esat M T, Fewell M P, Baxter A M and Burnett S M 1986 Nucl. Phys. A458165 Hague A M I, Casten R F, Forster I, Gelberg A, Rascher R, Richter A, yon Brentano P, Berreau G, Borner H G, Kerr S A, Schreckenbach K and Warner D D 1986 Nucl. Phys. A455 231 Halse P, Jacqua L and Barrett B R 1989 Phys. Rev. C40 968 Hartmann U, Bhole D, Guhr T, Hummel K D, Kilgus G, Milkau U and Richter A 1987 Nucl. Phys. A465 25 Hecht K T 1964 in Selected topics in nuclear spectroscopy (Compiler) B J Verhaar (Amsterdam: North Holland) pp. 51 Hecht K T 1965 Nucl. Phys. 62 1 Heyde K, Van Isacker P, Waroquier M, Wenes G, Gigase Y and Stachel J 1983 Nucl. Phys. A398 235 Hinds S, Bjerregaard J H, Hansen O and Nathan O 1965 Phys. Lett. 14 48 Iachello F (editor) •979 Interacting bosons in nuclear physics (New York: Plenum) Iachello F (editor) 1981 Interacting Bose-Fermi systems in nuclei (New York: Plenum Iachello F 1991 Nucl. Phys. A522 83c Iachello F and Arima A 1987 The interacting boson model (Cambridge: Univ. Press) sdg Interacting boson model 489 Iachello F and Talmi I 1987 Rev. Mod. Phys. 59 339 Iachello F and Van Isacker P 1991 The interacting boson fermion model (Cambridge: Univ. Press) Ichihara T, Sakaguchi H, Nakamura M, Yosoi M, Yeiri M, Tekeuchi Y, Togawa H, Tsutsumi T and Kobayashi S 1987 Phys. Rev. C36 1754 Jain A K, Sheline R K, Sood P C and Jain K 1990 Rev. Mod. Phys. 62 393 Jammari M K and Piepenbring R 1990 Nucl. Phys. AS10 339 Janecke J 1981 Phys. Lett. BI03 1 Janssens R V F and Khoo T L 1991 Ann. Rev. Nucl. Part. Sci. 41 321 Kalvius G M and Shenoy G K 1974 Nucl. Data Tables 14 639 Klein A and Marshalek E R 199t Rev. Mod. Phys. 63 375 Koike M, Nonaka I, Kokame J, Kamitsubo H, Awaya Y, Wada T and Nakamura H 1969 Nucl. Phys. A125 161 Kota V K B 1977 J. Phys. A10 L39 Kota V K B 1981 Ann. Phys. (NY) 134 221 Kota V K B 1982 SU(3) bands with g-bosons in IBA (University of Rochester Report UR-811) Kota V K B 1984 in Interacting boson-boson and boson-fermion systems (ed.) O Scholten (Singapore: World Scientific) pp. 83 Kota V K B 1986 Phys. Rev. C33 2218 Kota V K B 1987 in Recent trends in theoretical nuclear physics (ed.) K Srinivasa Rao (Delhi: Macmillan India Ltd) pp. 70, 102 Kota V K B, De Meyer H, Vander Jeugt J and Vanden Berghe G 1987 J. Math. Phys. 28 1644 Kota V K B and Parikh J C 1991 Group theoretical approach to nuclear collective motion (Lecture Notes) (Physical Research Laboratory report PRL-TN-91-73) Kota V K B and Potbhare V 1980 Phys. Rev. C21 2637 Kugel H W, Borchers R R and Kalish R 1972 Nucl. Phys. A186 513 Kusnezov D F 1988 Nuclear collective quadrupole-octupole excitations in the U(16) spdf interacting boson model Ph.D. "l-hesis (Princeton University) Kusnezov D F 1990 d. Phys. A23 5673 Kuyucak S, Lac V S and Morrison I 1991a Phys. Lett. B263 146 Kuyucak S, Lac V S, Morrison I and Barrett B R 1991b Phys. Lett. B263 347 Kuyucak S and Morrison I 1987a Phys. Rer. Lett. 58 315 Kuyucak S and Morrison I 1987b Phys. Rev. C36 774 Kuyucak S and Morrison I 1988a Ann. Phys. (NY) lgl 79 Kuyucak S and Morrison I 1988b Phys. Rev. C38 2482 Kuyucak S and Morrison I 1988c Phys. Lett. B202 169 Kuyucak S and Morrison I 1989a Phys. Rev. Lett. 62 1029 Kuyucak S and Morrison I 1989b Ann. Phys. (NY) 195 126 Kuyucak S and Morrison I 1990 Phys. Rev. C41 1803 Kuyucak S and Morrison I 1991 Phys. Lett. B255 305 Kuyucak S, Morrison I, Faessler A and Gelberg A 1988 Phys. Lett. B215 10 Kuyucak S, Morrison I and Sebe T 1991c Phys. Rev. C43 t187 Lange J, Kumar K and Hamilton J H 1982 Rev. Mod. Phys. 54 119 Leander G A and Chen Y S 1988 Phys. Rev. C37 2744 Leander G A and Sheline R K 1984 Nucl. Phys. A413 375 Lederer C M and Shirley V S 1987 Table of isotopes 7th edition (New York: John Wiley and sons) Lee P L and Boehm F 1971 Phys. Lett. B35 33 Lee I Y, Saladin J X, Baktash C, Holden J E and O'Brien J 1974 Phys. Rev. Lett. 33 383 Libert J and Quentin P 1982 Phys. Rev. C25 571 Ling Y 1983 Chin. Phys. (printed by APS) 3 87 Lipas P O, Toivonen P and Hammaren E 1987 Nucl. Phys. A469 348 Lipas P O, yon Bretano P and Gelberg A 1990 Rep. Prog. Phys. 53 1355 Lo Iudice N and Palumbo F 1978 Phys. Rev. Lett. 41 1532 Lo Iudice N and Palumbo F 1979 Nucl. Phys. A326 193 Maino G, Mengoni A and Ventura A 1990 Phys. Rev. C42 988 Matsuki S, Ohsawa T, Okamura H, Ogino K, Haga K, Sakomota N, Matoba M and Komatuzaki T 1986 J. Phys. Soc. Jpn. 55 Suppl p. 608 Mauthofer A, Stelzer K, Idzko J, Elze Th W, Wollersheim H J, Emling H, Fuchs P, Grosse E and Schwalm D 1990 Z. Phys. A336 263 490 Y Durga Devi and V Krishna Brahmam Kota Maxwell J R, Reynolds G M and Hintz N M 1966 Phys. Rev. 151 1000 Meyer H De, Van der Jeugt J, Vanden Berghe G and Kota V K B 1986 J. Phys. A19 L565 Miura K, Hiratate Y, Shoji T, Suehiro T, Yamaguchi H and Ishizaki Y 1985 Nucl. Phys. A436 221 Mizusaki T, Otsuka T and Sugita M 1991 Phys. Rev. C44 R1277 Mizusaki T, Yoshinaga N, Shigehara T and Cheon T 1991 Phys. Lett. B269 6 Morrison I 1986a Phys. Lett. B175 1 Morrison I 1986b J. Phys. G12 L201 Morrison I, Pieruschka P W and Wybourne B G 1991 J. Math. Phys. 32 356 Morrison I, von Brentano P and Gelberg A 1989 J. Phys. GI$ 801 Navratil P and Dobes J 1990 Nucl. Phys. A507 340 Navratil P and Dobes J 1991 Nucl. Phys. A532 223 Nazarewicz W and Rozmej P 1981 Nucl. Phys. A369 396 Nesterenko V O, Kukhitana I N, Sushkov A V and Khoa D T 1988 J. Phys. G14 723 Nesterenko V O, Solov'ev V G, Sushkhov A V and Shirikova N Yu 1986 Soy. J. Nucl. Phys. 44 938 Nilsson S G, Tsang C F, Sobiczewski A, Szymanski Z, Wycech S, Gustafson C, Lamm I L, Moiler P and Nilsson B 1969 Nucl. Phys. A131 1 Nuclear Data Sheets 1979 28 485 Nuclear Data Sheets 1986 48 345 Nuclear Data Sheets 1990 59 393 Okai T, Otsuka T and Arima A 1990 Book of absti'acts (Symposium in Honour of A. Arima, Santa Fe, New Mexico) pp. 73 Otsuka T 1981a Nucl. Phys. A368 244 Otsuka T 1981b Phys. Rev. Lett. 46 710 Otsuka T, Arima A and Iachello F 1978 Nucl. Phys. A309 1 Otsuka T, Arima A and Yoshinaga N 1982 Phys. Rev. Lett. 48 387 Otsuka T and Ginocchio J N 1985 Phys. Rev. Lett. 55 276 Otsuka T and Sugita M 1988 Phys. Lett. B215 205 Pan F, Cao Y F and Pan Z Y 1991 Chin. Phys. Lett. $ 5 Pannert W, Ring P and Gambhir Y K 1985 Nucl. Phys. A443 189 Piepenbring R and Jammari M K 1988 Nucl. Phys. A481 81 Pittel S, Dukelsky J, Perazzo R P J and Sofia H M 1984 Phys. Lett. B144 145 Powers R J, Barreau P, Bihoreau B, Miller J, Morgenstern J, Picard J and Roursel L 1979 Nucl. Phys. A316 295 Raduta A A and Lo Iudice N 1989 Z. Phys. A334 403 Raduta A A and De Lion D S 1990 Nucl. Phys. A513 11 Ragnarsson I and Broglia R A 1976 Nucl. Phys. A263 315 Rainwater J 1950 Phys. Rev. 79 432 Raman S, Malarkey C H, Milner W T, Nestor C W Jr., and Stelson P H 1987 At. Data Nucl. Data Tables 36 1 Ratna Raju R D 1981 Phys. Rev. C23 518 Ratna Raju R D 1986 J. Phys. GI2 L279 Ratna Raju R D, Draayer J P and Hecht K T 1973 Nucl. Phys. A202 433 Reuter W, Shera E B, Hoehn M V, Hersman F W, Milliman T, Finn J N, Hyde-Wright C, Lourie R, Pugh B and Bertozzi W 1984 Phys. Rev. C30 1465 Richter A 1988 Contemporary topics in nuclear structure physics (eds) R F Casten, A Frank, M Moshinsky and S Pittel (Singapore: World Scientific) pp. 127 Richter A 1991 Nucl. Phys. A522 139c Rohozinski S G 1988 Rep. Prog. Phys. 51 541 Rohozinski S G and Sobiczewski A 1981 Acta. Phys. Pol. B12 1001 Rowe D J 1985 Rep. Prog. Phys. 48 1419 Sage K A and Barrett B R 1980 Phys. Rev. C22 1765 Scholten O 1980 The interacting boson approximation model and applications Ph.D. thesis (University of Gr6ningen) Scholten O 1983 Phys. Rev. C28 1783 Scholten O, lachello F and Arima A 1978 Ann. Phys. (NY) 115 325 Scholten O and Warner D D 1984 Phys. Lett. B142 315 Sen X Y and Yu Z R 1987 J. Math. Phys. 28 2192 Sethi A, Hintz N M, Mihailidis D N, Mack A M, Gazzaly M, Jones K W, Pauletta G, Santi L and Goutte D 1991 Phys. Rev. C44 700 Sethi A, Todd Baker F, Emery G T, Jones W P and Grimm M A 1990 Nucl. Phys. A518 536 sd 9 Interacting boson model 491 Shaw A H and Greenberg J S 1974 Phys. Rev. CI0 263 Sirota S, Duarte J L M, Horodynski-Matsushigue L B, and Borello-Lewin T 1989 Phys. Rev. C40 1527 Solov'ev V G 1991 Capture gamma ray spectroscopy (ed.) R W Hoff(New York: AIP Conference Proceedings 238) pp. 16 Solov~ev V G and Shirikova N Yu 1981 Z. Phys. A301 263 Solov'ev V G and Shirikova N Yu 1989 Z. Phys. A334 149 Sood P C, Headly D M and Sheline R K 1991 At. Data. Nucl. Data Tables 47 89 Spear R H, Warner J P, Baxter A M, Esat M T, Fewell M P, Hinds S, Joye A M R and Kean D C 1977 Aust. J. Phys. 30 133 Stachel J, Van Isacker P and Heyde K 1982 Phys. Rev. C25 650 Suguna M, Ratna Raju R D and Kota V K B 1981 Pramana-J. Phys. 17 381 Sun H Z, Moshinsky M, Frank A and Van Isacker P 1983 Kinam 5 135 Svenne J P 1979 Adv. Nucl. Phys. 11 179 Todd Baker F 1985 Phys. Rev. C32 1430 Todd Baker F, Grimm M A, Scott A, Styles R C, Kruse T H, Jones K and Suchannek R 1981 Nucl. Phys. A371 68 Todd Baker F, Sethi A, Penumetcha V, Emery G T, Jones W P, Grimm M A and Whiten M L 1985 Phys. Rev. C32 2212 Todd Baker F, Sethi A, Penumetcha V, Emery G T, Jones W P, Grimm M A and Written M L 1989 Nucl. Phys. AS01 546 Twin P J 1988 Contemporary topics in nuclear structure physics (ed.) R F Casten, A Frank, M Moshinsky and S Pittel (Singapore: World Scientific) pp. 445 Van Egmond A and Allaart K 1983 Phys. Lett. B131 275 Van Egmond A and Allaart K 1984 Nucl. Phys. A425 275 Van Isacker P and Frank A 1989 Phys. Lett. B225 1 Van Isacker P, Heyde K, Waroquier M and Wenes G 1981 Phys. Lett. B104 5 Van Isacker P, Heyde K, Waroquier M and Wenes G 1982 Nucl. Phys. A380 383 Van Isacker P, Heyde K, Jolie J and Sevrin A 1986 Ann. Phys. (NY) 171 253 Vanthournout J, De Meyer H and Vanden Berghe G 1988 J. Math. Phys. 29 1958 Vanthournout J, De Meyer H and Vanden Berghe G 1989 J. Math. Phys. 30 943 Vervier J 1987 Rivista Del Nuovo Cimento 10 No. 9 Walker P M, Carvalho J L S and Bernthal F M 1982 Phys. Lett. B168 393 Wapstra A H and Bos K 1977 At. Data Nucl. Data Tables 19 175 Warner D D and Van Isacker P 1990 Phys. Lett. B247 1 Wesseling J, De Jager C W, Vanderlaan J B, Devries H, Harakeh M N 1991 Nucl. Phys. A535 285 Whitehead R R, Watt A, Cole B J and Morrison I 1977 Adv. Nucl. Phys. 9 123 Wu H C 1982 Phys. Lett. Bll0 1 Wu H C, Dieperink A E L and Pittel S 1986 Phys. Rev. C34 703 Wu H C, Dieperink A E L and Scholton O 1987 Phys. Lett. B187 205 Wu H C, Dieperink A E L, Scholten O, Harakeh M N, De Leo R, Pignanelli M and Morrison I 1988 Phys. Rev. C38 1638 Wu H C and Zhou X Q 1984 NucL Phys. A417 67 Xing Z and Xie H X 1990 Chin. Phys. (printed by APS) 10 375 Yoshinaga N 1986 Nucl. Phys. A456 21 Yoshinaga N 1989a Nucl. Phys. A493 323 Yoshinaga N 1989b Nucl. Phys. A503 65 Yoshinaga N 1991 Nucl. Phys. A522 99c Yoshinaga N, Akiyama Y and Arima A 1986 Phys. Rev. Lett. 56 1116 Yoshinaga N, Akiyama Y and Arima A 1988 Phys. Rev. C38 419 Yoshinaga N, Arima A and Otsuka T 1984 Phys. Lett. B143 5 Yoshinaga N and Brink D M 1990 Nucl. Phys. A515 1 Yu Z 1988 Mod. Phys. Lett. A3 1059 Yu Z R, Scholten O and Sun H Z 1986 d. Math. Phys. 27 442 Ziegler W, Rangacharyulu C, Richter A and Spieler C 1990 Phys. Rev. Lett. 65 2515 Zuffi L, Maino G and Ventura A 1986 Phys. Rev. C34 1223 Zumbro J D, Naumann E B, Hoehn M V, Reuter W, Shera E B, Bemis C E Jr and Tanaka Y 1986 Phys. Lett. B167 383 Zumbro J D, Shera E B, Tanaka Y, Bemis C E Jr, Naumann R A, Hoehn M V, Reuter W and Steffen R M 1984 Phys. Rev. Lett. 53 1888